Unified List of Undergraduate Courses

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Compulsory Courses

Infinitesimal Calculus I (MAY111)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAY111
Semester	1
Course Title	Infinitesimal Calculus I
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 5, 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 (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Here, the acronym RFooV stands for Real Function of one Variable. Remembering: Introduction to the sets of Natural Numbers, Integer Numbers, Rational Numbers, Irrational Numbers and Real Numbers, viewed from the aspect of Mathematical Analysis. Bounded and not bounded subsets of such sets. Basic concepts of Trigonometry. The concept of RFooV. Some basic properties of such functions. Elementary RFooVs. The concept of real valued sequences. Study of such sequences, including existence and calculation of limits. Limits and continuity of RFooV, using the (ε-δ) definition and the sequential definition. Basic properties of convergent RFooVs. Basic properties of continuous RFooVs. Classes of non-continuous RFooVs. Derivative of RFooV using the (ε-δ) definition and the sequential definition. Derivatives of elementary RFooVs. Calculation of derivatives. Comprehension: Methods of establishing a mathematical concept based on axioms and based of construction. Calculation and finding properties of sets of real numbers. Minimum upper and maximum lower boundaries. Graphing RFooVs, monotone RFooVs, bounded RFooVs, periodic RFooVs. Subsequences, the Bolzano-Weierstass Theorem, Cauchy sequences. Local behaviour of continuous RFooVs. The Bolzano Theorem and the Intermediate Values Theorem. Properties of continuous RFooVs defined in closed intervals, continuity of reverse continuous RFooVs. Uniform continuity of RFooVs defined in closed intervals. Methods of derivation, higher order derivatives. The Rolle Theorem, the Mean Value Theorem, the Darboux Theorem. The connection between derivative and monotonicity, extrema of RFooVs, convex and concave RFooVs, inflections points. Theorems for the derivation of inverse RFooVs. Generalized Mean Value Theorem, the De L’ Hospital Rule. Studying RFooVs using derivatives. Applying: Existence and uniqueness of solutions of non-linear equations. Finding maximum and minimum values of quantities, which emerge in problems in Natural Sciences. Plotting RFooVs. Evaluating: Teaching undergraduate courses.

General Competences

Creative, analytical and inductive thinking. Required for the creation of new scientific ideas. Working independently. Working in groups. Decision making.

Syllabus

• Real numbers, axiomatic foundation of the set of real numbers (emphasis in the notion of supremum and infimim), natural numbers, induction, classical inequalities.
• Functions, graph of a function, monotone functions, bounded functions, periodic functions. Injective and surjective functions, inverse of a function. Trigonometric functions, inverse trigonometric functions, exponential and logarithmic functions, hyperbolic and inverse hyperbolic functions.
• Sequences of real numbers, convergent sequences, monotone sequences, sequences defined by recursion, limits of monotone sequences, nested intervals. The notion of subsequence, Bolzano Weierstass’ Theorem, Cauchy sequences. Accumulation points of sequences, upper and lower limit of a sequence (limsup, liminf).
• Continuity of functions, accumulation points and isolated points, limits of functions, one sided limits, limits on plus infinity and minus infinity. Continuity of several basic functions, local behaviour of a continuous function. Bolzano Theorem and intermediate value theorem. Characterization of continuity via sequences, properties of continuous functions defined on closed intervals, continuity of inverse functions.
• Derivative of a function, definition and geometric interpretation, examples and applications in sciences. The derivatives of elementary functions, derivation rules, higher order derivation. Rolle’s Theorem, Mean Value Theorem, Darboux’s theorem. Derivative and the monotonicity of a function, extrema of functions, convex and concave functions, inflection points. Derivation of inverse functions. Generalized Mean Value Theorem, De L’ Hospital rule. Study of functions using derivatives.

Teaching and Learning Methods - Evaluation

Delivery	Lectures in class. Learning Management System (e.g.: Moodle).
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 65 Study and analysis of bibliography 100 Preparation of assignments and interactive teaching 22.5 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

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Fundamental Concepts of Mathematics (MAY112)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAY112
Semester	1
Course Title	Fundamental Concepts of Mathematics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 5, 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

As a first step, the students get familiar with basic tools of logic, set theory (set operations and properties), relations and functions. Emphasis is given to notions such as collections and families (coverings) bounds (max, min, sup, inf) as well as to images and pre-images of sets under functions. Part of the kernel of the course is a detailed axiomatic construction of the real numbers aiming that the students acknowledge this set as result of an axiomatic construction rather than of an empiric approach, yet the value and the significancy of the axiomatic foundation of mathematical structures be apparent. In the section concerning cardinality of sets, besides arithmetic of finite sets, students classify types of infinite sets (finite, numerable, denumerable) and approach in an abstract way the notion of infinity in relation with sets in common use as the sets of naturals, integers, rationals, and reals. A major course learning outcome is that assimilation of the offered knowledge will create a good qualitative background so that students be able to proceed with adequacy to studying other branches of mathematics.

General Competences

Analysis and synthesis of data and information Individual work Team work Production of creative and inductive thinking Production of analytical and synthetic thinking

Syllabus

Definition of trigonometric numbers, trigonometric cycle. Trigonometric numbers of the sum of two angles and trigonometric numbers of the double of an arc. Trigonometrical functions. Trigonometrical equations. Transformations of products to sum and of sums to products.
Elements of Logic. Basic set theory, operations and properties, power set, Cartesian products, collections. Relations, properties, equivalence relations, order relations, bounded sets, well ordered sets, principle of infinite reduction, functions, one to one functions, onto functions.
Image and preimage of a set, functions and ordered sets. Families. The set of real numbers: axiomatic approach. The sets of natural numbers, integers. The field of rational numbers. Roots of nonnegative real numbers. The set if irrational numbers.
The axiom of completeness and equivalent statements. Equivalent sets. Finite sets. Infinite sets. Schroder-Bernstein theorem. Numerable sets. At most numerable sets. Denumerable sets. Cantor’ theorem. Axiom of Choice and equivalent statements. A first approach to the necessity of an axiomatic foundation of sets.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	Use of ICT (Tex, Mathematica etc.) for presentation of essays and assignments.
Teaching Methods	Activity Semester Workload Lectures 65 Study and analysis of bibliography 22.5 Preparation of assignments and interactive teaching 100 Course total 187.5
Student Performance Evaluation	Written examination at the end of the semester including theory and problems-exercises.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• K. G. Binmore, Logic, Sets and Numbers, Cambridge University Press, 1980.
• W. W. Fairchild and C. I. Tulcea, Sets, W. B. Shaunders Co. Philadelphia, 1970.
• S. Lipschutz, Set Theory and Related Topics, Schaum’s Outline Series, New York, 1965.
• D. Van Dalen, H. C. Doets and H. Deswart, Sets: Naïve, Axiomatic and Applied, Pergamon Press, Oxford, 1987.

Linear Algebra I (MAY121)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAY121
Semester	1
Course Title	Linear Algebra I
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 5, 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	After finishing the course, the students will be able: to use matrices as a tool in theoretical or numerical computations to compute the rank of a matrix to compute determinants to solve linear systems of equations to understand and use the notion of vector space.
General Competences	The aim of the course is to empower the graduate to analyse and compose basic notions and knowledge of Linear Algebra and advance his creative and productive thinking.

Syllabus

• The algebra of (m x n) matrices and applications.
• Row echelon forms and reduced row echelon form of a matrix.
• Rank of a matrix. Determinants. Invertible matrices.
• Linear systems and applications.
• Vector spaces. Linear maps.
• The space L(E,F) of linear operations.
• Subspaces. Bases. Dimension. Rank of a linear operation.
• Fundamental equation of dimension and its applications. Matrix of a linear map. Matrix of a change of bases. The isomorphism between linear mapsand matrices. Equivalent matrices. Similar matrices. Determinant of an endomorphism. Sum and direct sum of vector subspaces.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Teaching Material: Teaching material in electronic form available at the home page of the course. Communication with the students: Office hours for the students (questions and problem solving). Email correspondence Weekly updates of the homepage of the course.
Teaching Methods	Activity Semester Workload Lectures (13x5) 65 Working independently 100 Exercises-Homeworks 22.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students, in English) which includes analysis of theoretical topics and resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Introduction to Linear Algebra (Greek), Bozapalidis Symeon, ISBN: 978-960-99293-5-6 (Editor): Charalambos Nik. Aivazis

Number Theory (MAY123)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAY123
Semester	1
Course Title	Number Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 4, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, English
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 the study of the structure and basic properties of natural numbers, and more generally of integers. This study is based on the fundamental concept of divisibility of integers, and the (unique) factorization of a natural number into prime factors. The most important ideas, concepts and results that allow us to understand the structure and fundamental properties of all positive integers with respect to divisibility, are as follows (Keywords of course): Divisibility, prime numbers, Euclidean algorithm, greatest common divisor and least common multiple. Congruences and systems of congruences, Chinese remainder theorem. Arithmetical functions and Moebius inversion formula. Euler’s φ-function. Theorems of Fermat, Euler and Wilson. Primitive mod p roots. Theory of indices and quadratic residues. Law of quadratic reciprocity. Applications to cryptosystems. We will formulate and prove several theorems concerning the structure of all integers through the concept of divisibility. During the course will analyse applications of Number Theory to other sciences, and particularly to Cryptography. This course is an introduction to the basic results, the basic methods, and the basic problems of elementary number theory, and requires no special knowledge of other subjects of the curriculum. At the end of the course we expect the student to (a) have understood the definitions and basic theorems concerning the divisibility structure of the integers which are discussed in the course, (b) to have understood how they are applied in discrete examples, (c) to be able to apply the material in order to extract new elementary conclusions, and finally (d) to perform some (no so obvious) calculations.

General Competences

The course aims to enable the undergraduate student to acquire the ability to analyse and synthesize basic knowledge of the theory of numbers, to apply basic examples in other areas, and in particular to solve concrete problems concerning properties of numbers occurring in everyday life. The contact of the undergraduate student with the ideas and concepts of number theory, (a) promotes the creative, analytical and deductive thinking and the ability to work independently, (b) improves his critical thinking and his ability to apply abstract knowledge in various field.

Syllabus

• Complex numbers.
• Divisibility.
• Congruences mod m.
• Chinese remainder theorem.
• Arithmetical functions and Moebius inversion formula.
• The theorems of Fermat, Euler and Wilson.
• Primitive roots mod p.
• The theory of indices and the Law of quadratic reciprocity.
• Applications to cryptography.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Teaching Material: Teaching material in electronic form available at the home page of the course. Communication with the students: Office hours for the students (questions and problem solving). Email correspondence Weekly updates of the homepage of the course.
Teaching Methods	Activity Semester Workload Lectures 52 Working independently 104 Exercises-Homeworks 31.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students, in English) which includes analysis of theoretical topics and resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Infinitesimal Calculus II (MAY211)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAY211
Semester	2
Course Title	Infinitesimal Calculus II
Independent Teaching Activities	Lectures, laboratory exercises (Weekly Teaching Hours: 5, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	None (from the typical point of view). Without the knowledge earned from the course “Infinitesimal Calculus I” will be nearly impossible to follow this course.
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (exams in English are provided for foreign students)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

This course is the sequel of the course “Infinitesimal Calculus I”. The student will get in contact with more notions and techniques in the branch of Analysis. In this course the students: Are taught the notions of convergence and absolute convergence of series. They learn criteria and theorems concerning these notions as well as they learn how to compute sums of series. They are introduced in the notion of power series and they learn how to calculate the radius of convergence of a power series. Are taught the notion of uniform continuity and they learn to distinguish this notion from continuity. Are taught the notion of Riemann integral and various theorems concerning this notion. They also learn various integrating techniques. Are taught Taylor’s theorem and they learn to write a given function as a Taylor series.

General Competences

The course provides inductive and analytical thinking, the students evolve their computational skills and they get knowledge necessary for other courses during their undergraduate studies.

Syllabus

Series, convergence of series and criteria for convergence of series. Dirichlet’s criterion, D’ Alembert’s criterion, Cauchy’s criterion, integral criterion. Series with alternating signs and Leibnitz’s theorem. Absolute convergence and reordering of series, Power series, radius of convergence of power series.
Uniform continuity, definition and properties. Characterization of uniform continuity via sequences. Uniform continuity of continuous functions defined on closed intervals.
Riemann integral, definition for bounded functions defined on closed intervals. Riemann’s criterion, integrability of continuous functions. Indefinite integral and the Fundamental theorem of Calculus. Mean Value theorem of integral calculus, integration by parts, integration by substitution. Integrals of basic functions, integrations of rational functions. Applications of integrals, generalized integrals, relation between generalized integrals and series.
Taylor polynomials, Taylor’s Theorem, forms of the Taylor remainder. Taylor series and expansions of some basic functions as Taylor series.

Teaching and Learning Methods - Evaluation

Delivery	Due to the theoretical nature of this course the teaching is exclusively given in the blackboard by the teacher.
Use of Information and Communications Technology	The students may contact their teachers by electronic means, i.e. by e-mail.
Teaching Methods	Activity Semester Workload Lectures (13x5) 65 Solutions of exercises 22.5 Individual study 100 Course total 187.5
Student Performance Evaluation	Exams in the end of the semester (mandatory). Assignments of exercises during the semester (optional).

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Thomas, Απειροστικός Λογισμός, R.L. Finney, M.D. Weir, F.R.Giordano, Πανεπιστημιακές Εκδόσεις Κρήτης, (Απόδοση στα ελληνικά: Μ. Αντωνογιαννάκης).

Linear Algebra II (MAY221)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAY221
Semester	2
Course Title	Linear Algebra II
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 5, 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	After finishing the course, the students will be able: to compute eigenvalues and eigenvectors to diagonalize matrices to compute othocanonical bases, orthogonal complements and orthogonal projections to subspaces to diagonalise symmetric matrices using orthogonal matrices to compute the invariants of quadratic forms.
General Competences	The aim of the course is to empower the graduate to analyse and compose notions and knowledge of Linear Algebra and advance creative and productive thinking.

Syllabus

Eigenvalues, Eigenvectors, Eigenspaces, Diagonalisation, Cauley-Hamilton thoerem, Euclidean spaces, Orthogonality, Gram-Schmidt orthogonalization, Orthogonal matrices, Self-adjoint endomorphisms, Symmetric matrices, Spectral theorem, Isometries, Quadratic forms, Principal Axes, Square root of a nonnegative real symmetric matrix. Norms of a matrix.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures Lectures (13X5) 65 Working independently 100 Exercises-Homeworks 22.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which includes resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Introduction to Linear Algebra (Greek), Bozapalidis Symeon, ISBN: 978-960-99293-5-6 (Editor): Charalambos Nik. Aivazis

Analytic Geometry (MAY223)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAY223
Semester	2
Course Title	Analytic Geometry
Independent Teaching Activities	Lectures, laboratory exercises (Weekly Teaching Hours: 5, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, English
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

It is an introductory course on geometry. The aim is to study problems in geometry using rectangular coordinates and tools based on Linear Algebra. On completion of the course the student should be familiar with basic notions in geometry like the one of isometry. Furthermore, the student should have a background to allow him to attain more advanced courses on geometry, calculus of several variables and others.

General Competences

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

Syllabus

Axioms of Euclidean geometry (plane and space) and proofs of basic propositions. Cartesian model, vectors, linear independence, bases, coordinates and applications. Inner product, cross product, area, volume and determinants. Lines and planes. Geometric transformations (parallel transports, rotations, reflections), isometries and the notion of congruence. Transformation of area and volume under linear transformations. Curves and surfaces of 2nd degree and their classification. Curves, surfaces and parametrizations.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures (13X5) 65 Working independently 100 Exercises-Homeworks 22.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which includes resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Introduction to Computer Science (MAY242)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAY242
Semester	2
Course Title	Introduction to Computer Science
Independent Teaching Activities	Lectures and laboratory exercises (Weekly Teaching Hours: 5, 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

This course offers an introduction to the Computer Science. It mainly focuses on how to algorithmically solve simple and complex mathematical problems. It provides basic programming techniques using a high-level programming language such as C/C ++. Moreover, the course analyzes the basic numbering systems, it provides the basic arithmetic operations in different numerical systems and refers to the representation of information on computer systems. Additionally, the course provides basic concepts of mathematical logic, such as Boolean algebra, and principles that govern the semantic and syntactic approach of propositional logic. Upon completion of the course, the students will be able to: Recognize different numbering systems and process number representations on computer systems. Understand basic concepts and theorems of propositional logic, make use of metatheorems of propositional logic and understand the formal proof of logical expressions through the syntactic approach. Describe basic algorithms for solving simple and complex mathematical problems and implement algorithms by using basic concepts of a programming language (C/C ++). The course includes laboratory exercises in which the participation is obligatory.

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 Development of algorithmic thinking

Syllabus

• Introduction to Numerical Representation
  1. Arithmetic operations in numerical systems
  2. Representations of binary numbers
• Introduction to Mathematical Logic (Boolean Algebra)
  1. Semantic approach: principles of propositional logic, conjunctive normal form (CNF), complete sets, meta-theorems
  2. Syntactic approach: axioms, Modus Ponens rule, meta-theorems (abduction, inversion), validity and completeness theorems.
• Basic Programming Techniques with programming language C/C++
  1. Input/Output data, type of structures and variables
  2. Flow control if/else
  3. Loop structures: for, while, do-while
  4. Defensive Programming
  5. Arrays (one dimension and multidimensions)

Teaching and Learning Methods - Evaluation

Delivery	Lectures, labs session
Use of Information and Communications Technology	Projector and interactive board during lectures. Computer for demonstration of programming. Computers in laboratories for development and testing of programs. Course website maintenance. Announcements and posting of teaching material (lecture slides and notes, programs). Assessment marks via the ecourse platform by UOI.
Teaching Methods	Activity Semester Workload Lectures (13X5) 65 Self study 100 Exercises 22.5 Course total 187.5
Student Performance Evaluation	Written final exam (70%) Multiple choice questions. Develop programs and Implementation. Laboratory exercises (30%). Multiple choice questions. Develop programs and Implementation.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Η. Deitel and P. Deitel, C++ Προγραμματισμός 6η Εκδοση, Εκδόσεις Μ. Γκιούρδας, 2013. Κωδικός Ευδ: 12536819.
• Κωδικός Ευδόξου [77106820]: Διακριτά μαθηματικά και εφαρμογές τους, 8η Έκδοση, Kenneth H. Rosen
• Κωδικός Ευδόξου [86055409]: Διακριτά μαθηματικά, Hunter David (Συγγρ.)
• Κωδικός Ευδόξου [77109607]: Εισαγωγή στην πληροφορική, Evans Alan, Martin Kendall, Poatsy Mary Anne.
• Ζάχος, Ε., Παγουρτζής, Α., Σούλιου, Θ., 2015. Θεμελίωση επιστήμης υπολογιστών. [ηλεκτρ. βιβλ.] Αθήνα:Σύνδεσμος Ελληνικών Ακαδημαϊκών Βιβλιοθηκών. Διαθέσιμο στο: http://hdl.handle.net/11419/545
• [Περιοδικό / Journal] IEEE Transactions on Computers

Infinitesimal Calculus III (MAY311)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAΥ311
Semester	3
Course Title	Infinitesimal Calculus III
Independent Teaching Activities	Lectures, laboratory exercises (Weekly Teaching Hours: 5, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, English
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 learning outcomes are the: differentiability analysis of real- and vector-valued functions of several variables familiarity with the Euclidean space from an analytic (topological) viewpoint knowledge of the problems that arise in Analysis in several dimensions preparation for the treatment of functions of several variables in more specialized courses, e.g., Partial Differential Equations, Differential Geometry, Classical Mechanics, Application of Mathematics in the Sciences development of combination skills concerning knowledge from diverse areas of Mathematics (Linear Algebra, Analytical Geometry, Analysis).

General Competences

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

Syllabus

• Algebraic and topological structure of the Euclidean space R^n and geometric representation of the two- and three-dimensional space. Vector-sequences and their use concerning the topology of R^n.
• Real- and Vector-valued functions of several variables. Limits and continuity of functions.
• Partial derivatives. Partially differentiable and differentiable functions. Directional derivative. Differential operators and curves in R^n.
• Higher order partial derivatives. Taylor Theorem. Local and global extrema of real-valued functions. Implicit Function Theorem. Inverse Function Theorem. Constrained extrema.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Teaching material is offered at the course's website (notes and older exams) The students may contact the lecturer by e-mail
Teaching Methods	Activity Semester Workload Lectures (13X5) 65 Working independently 100 Exercises-Homeworks 22.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Giannoulis, I. (2015), Διανυσματική ανάλυση, Kallipos, Open Academic Editions

Introduction to Probability (MAY331)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΥ331
Semester	3
Course Title	Introduction to Probability
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 5, 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, reading Course)
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 to provide with a comprehensive understanding of the basic definitions of probability and the basic principles and laws of probability theory. Further, the introduction to the concepts of the random variable and the distribution function, as well as, their characteristics, such as the mean, variance, moments, moment generating function, etc., is included in the main aims of the course. Special distributions, such as binomial, geometric, Pascal, Poisson, uniform, exponential, gamma, normal distribution, etc. are studied and their use and application is indicated. The course is compulsory, it is of an entry-level and it aims to develop skills that help the students to understand, design and exploit stochastic models to describe real problems. At the end of the course the students is expected to be able to: Exploit and apply the classical and empirical definition of probability in order to calculate probabilities, by using combinatorial analysis. Utilize the axiomatic foundation of the concept of probability and use it in order to derive and prove probabilistic laws and properties. Understand and utilize classical probabilistic laws as the multiplicative theorem, the total probability theorem, Bayes’ formula, and independence for modeling respective problems. Emphasis is given to the use of interdisciplinary problems which are modeled by the application of the above probabilistic rules. Understand the necessity of introducing and studying the concept of random variable, its characteristics (mean, variance, etc.) and the corresponding probability distribution. Special discrete and continuous distributions are defined and utilized for the description, analysis and study real problems from different areas (lifetime distributions, reliability etc.).

General Competences

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

Syllabus

Basic ideas and laws of probability: Sample space and events. Classical-Statistical and Axiomatic definition of probability. Properties of probability and probabilistic formulas and laws. Elements of combinatorial analysis. Random variables and distribution functions. Discrete and continuous random variables and distribution functions. Standard discrete and continuous distributions: Binomial, Geometric, Pescal, Poisson, Uniform, Exponential, gamma, Normal etc. Characteristics of random variables and probability distributions: Expectation, variance, moments, moment generating function, properties. Transformation of random variables.

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 (13X5) 65 Working independently 100 Exercises-Homeworks 22.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which concentrates on the solution of problems which are motivated by the main themes of the course.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Ι. Κοντογιάννης, Σ. Τουμπής. Στοιχεία πιθανοτήτων, [Προπτυχιακό εγχειρίδιο]. Κάλλιπος, Ανοικτές Ακαδημαϊκές Εκδόσεις. https://hdl.handle.net/11419/2810.
• J. Blitzstein, J. Hwang. Introduction to Probability, 2nd edition, CRC Press, 2019.
• R. Dobrow. Probability with Applications and R, Wiley, 2014.
• H. Tijms. Understanding Probability, 3rd edition, Cambridge University Press, 2012.
• H. Tijms. ProbabilityQ a lively introduction, Cambridge University Press, 2018.
• [Περιοδικό / Journal] Annals of Probability (IMS)
• [Περιοδικό / Journal] Electronic Journal of Probability (IMS)
• [Περιοδικό / Journal] Journal of Applied Probability (Cambridge University Press)

Introduction to Numerical Analysis (MAY341)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑY341
Semester	3
Course Title	Introduction to Numerical Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 4, 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

Upon successful completion of this course, students will be able to: recognise key numerical methods from a variety of maths problems and apply them for the solution of actual problems. apply a variety of theoretical techniques for deriving and analyzing the error of numerical approximations. analyse and evaluate the accuracy of common numerical methods. evaluate the performance of numerical methods in terms of accuracy, efficacy, and applicability.

General Competences

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

Syllabus

• Error Analysis.
• Numerical solution of nonlinear equations: iterative methods, the fixed-point theorem, Newton’s method, the secant method.
• Numerical solution of linear systems: Matrix norms and conditioning. Direct Methods (Gauss elimination, LU factorization). Iterative methods, convergence, and examples of iterative methods (Jacobi, Gauss-Seidel).
• Polynomial interpolation: Lagrange and Hermite interpolation. Linear splines. Error analysis of interpolation.
• Numerical integration: Newton-Cotes quadrature formula (the trapezoidal rule and Simpson’s rule). Error analysis of numerical integration.

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 (13X4) 52 Study and analysis of bibliography 100 Exercises-online Quizzes 35.5 Course total 187.5
Student Performance Evaluation	Written examination (Weighting 100%, addressing learning outcomes 1-4)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• “An Introduction to Numerical Analysis”, E. Süli, and D. Mayers, Cambridge University Press, Cambridge, 2003.

Introduction to Programming (MAY343)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAY343
Semester	3
Course Title	Introduction to Programming
Independent Teaching Activities	Lectures, laboratory exercises, tutorials, quiz (Weekly Teaching Hours: 5, 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

This course aims at analyzing and solving problems using the computer as well as at introducing a high-level programming language (which in this case is C++ and Python). After successfully passing this course, the students will be able to: Write simple or complex programs. Verify the correctness and appropriateness of a given program. Debug programs. Understand basic programming concepts, structures and techniques. Use arrays, strings, and functions. Understand elementary notions of object-oriented programming. Conduct simple and complex arithmetic computations via programming. Use control flow constructs, conditions, decision structures and loops. Structure their programs with the help of iterative and recursive functions. Program basic operations on data, such as searching and sorting.

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

• Introduction to programming
• Preprocessing, numerical, boolean and logical operators
• Flow control: if/else, switch, for, while, do-while
• Structuring, locality of parameters, pass by value/reference, variable scope, recursive functions, program stack.
• Arrays, strings, objects
• Input/Output
• Functions, variables’ scope and recursion
• Searching and sorting data
• Elementary data structures.

Teaching and Learning Methods - Evaluation

Delivery	Lectures, labs session
Use of Information and Communications Technology	Use of projector and interactive board during lectures. Use of computer for demonstation of programming. Use of computers in laboratories for development and testing of programs. Course website maintenance. Announcements and posting of teaching material (lecture slides and notes, programs). Announcement of assessment marks via the ecourse platform by UOI.
Teaching Methods	Activity Semester Workload Lectures (13X5) 65 Laboratory practice 100 Tutorials 22.5 Course total 187.5
Student Performance Evaluation	Final written examination (80%) Multiple choice questions Develop programs Laboratory exercises (20%) Multiple choice questions Develop programs

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• L. Jesse, Πλήρες εγχειρίδιο της C++, Εκδόσεις Α. Γκιούρδα, 2006. Κωδικός Ευδ: 12374.
• Βιβλίο [50656350]: Υπολογισμοί και Προγραμματισμός με την Python, John V. Guttag, Κλειδάριθμος, 2015.
• Βιβλίο [59357236]: Εισαγωγή στον Προγραμματισμό με την Python, Schneider David
• Βιβλίο [77119000]: Προγραμματισμός με την Python, Στράτος Καλαφατούδης, Γεώργιος Σταμούλης
• Βιβλίο [320152]: Εισαγωγή στον Προγραμματισμό με αρωγό τη γλώσσα Python [Ηλεκτρονικό Βιβλίο], Γεώργιος Μανής
• Βιβλίο [174838]: Python Scripting for Computational Science [electronic resource], Hans Petter Langtangen
• Βιβλίο [170352]: Beginning Python [electronic resource], Magnus Lie Hetland
• [Περιοδικό / Journal] Science of Computer Programming, ELSEVIER.
• [Περιοδικό / Journal] ACM Transactions on Programming Languages and Systems (TOPLAS)

Infinitesimal Calculus IV (MAY411)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAY411
Semester	4
Course Title	Infinitesimal Calculus IV
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 5, 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 (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Here, the acronym VFomV stands for Vector Function of multiple Variables. Remembering: The concept of the integral of VFomV. Basic properties of this integral. The concept of improper integral of VFomV. Basic properties of this integral. The concept of integrals of VFomV over paths and surfaces. Basic properties of this integral. The concepts of vector field and gradient field. The concepts of the sequence of VFomVs, of uniform convergence of such sequences, of the series of such sequences and of the Fourier series. Comprehension: Integration of VFomV over a rectangle and over an elementary region. Changing the order of integration. Integration over vector fields and gradient fields. The Stokes, Green and Gauss Theorems. Applying: Finding length of path, area of elementary region, volume of solid body. Finding curvature of surfaces and minimal surfaces. Conservative fields and their applications in Physics. Study of liquid fluids and study of waves. Differential forms and their applications in Differential Geometry. Evaluating: Teaching undergraduate and graduate courses.

General Competences

Creative, analytical and inductive thinking. Required for the creation of new scientific ideas. Working independently. Working in groups. Decision making.

Syllabus

Definition of multiple integral using lower and upper sums over closed rectangles, set of zero volume, Lebesgue Criterion for Riemann Integrability, Jordan measurable sets and the definition of the integral over such sets, Fubini Theorem, Cavalieri Principle, elementary regions in two and three dimensional spaces, change of variables and their basic applications, evaluation of integrals using the aforementioned methods. Definition of integrals over paths for parametrizes functions an vector fields, definition of path length, parametrizes paths, parametrized transformations, gradient fields and path independent integrals, Green Theorem. Surfaces and parametrization of surface integrals. Definition of surface integral for real functions and for vector fields. Area of surface. Stokes and Gauss Theorems. Uniform convergence of function’s sequences and series. Fourier series.

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 and Communication Platform, for distributing teaching material, submission of assignments, course announcements, gradebook keeping for all students evaluation procedures, communicating with students. Use of Web Appointment Scheduling System for organising office appointments. 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 (13X5) 65 Study and analysis of bibliography 100 Preparation of assignments and interactive teaching 22.5 Course total 187.5
Student Performance Evaluation	Language of evaluation: Greek and English. Methods of evaluation: Weekly written assignments. Few number of tests during the semester. Based on their grades in the aforementioned weekly assignments and tests, limited number of students can participate in exams towards the end of the semester, before the beginning of the exams period. In any case, all students can participate in written exams at the end of the semester, during the exams period. 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

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Metric Spaces and their Topology (MAY413)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAY413
Semester	4
Course Title	Metric Spaces and their Topology
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 5, 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

Topology is a powerful tool for research and expression in all branches of Mathematical Science. In the last few years, Topology has been increasingly used in the creation of mathematical models that serve research applied disciplines such as Economics, Meteorology, Insurance Mathematics, Epidemiology in Medicine, etc. The didactic approach here is to initially give the theory of metric spaces and then, as a mere reference, an introduction to General Topology. An in-depth study of Metric spaces, in addition to preparing the student to accept the abstract structures of General Topology, helps him to better understand the structure of the Euclidean space n, which is studied the same time in the Multi-Variable Infinitesimal Calculus. Topics which are covered are convergence, continuity, completeness, total boundness, compactness, separability and connectedness. These concepts, as well as the proofs of the related results, are given in such a way that, wherever possible, they can be easily and without major changes adapted toTopological Spaces.

General Competences

Analysis and synthesis of data and information Autonomous work Teamwork Working in an interdisciplinary environment Promoting creative and inductive thinking Promoting analytical and synthetic thinking Production of new research ideas.

Syllabus

Metric spaces, definition, examples, basic properties. Metrics in vector spaces induced by norms. Diameter of a set, distance of sets. Sequences in metric spaces, subsequences, convergence of sequences. Functions between metric spaces, continuous functions, characterization of continuity via sequences, uniform continuity of functions. Open balls, closed balls, interior, closed hull and boundary, accumulation points and derived set. The topology of a metric space, the concept of a topological space. Basic (or Cauchy) sequences, complete metric spaces. Principle of contraction (Banach's Fixed Point Theorem). Totally bounded metric spaces, compact spaces. Equivalent forms of compactness of metric spaces. Properties of compact spaces. Separable metric spaces. Connectedness in metric spaces, properties of connected sets, connected components.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	Use of ICT for presentation of essays and assignments
Teaching Methods	Activity Semester Workload Lectures 65 Solving exercises at home 22.5 Individual study 100 Course total 187.5
Student Performance Evaluation	Written examination at the end of the semester including theory and problems-exercises.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• K. W. Anderson and D. W. Hall, Sers, Sequences and Mappings, John Wiley and Sons, Inc. New York 1963.
• V. Arkhangel’skii and V.I. Ponomarev, Fundamentals of General topology, D. Reidel Publishing Company, 1983.
• G. Buskes and A. van Rooij, Topological Spaces, Springer-Verlag, New York, 1197.
• D. C. J. Burgess, Analytical Topology, D. Van Nostrand Co. Ltd., London, 1966.
• N. L. Carothers, Real Analysis, Cambridge University Press, 2000.
• E. Copson, Metric Spaces, Cambridge University Press, 1968.
• J. Diedonne, Foundations of Modern Analysis, Academic Press, New York, 1966.
• J. Dugudji, Topology, Allyn and Bacon Inc., Boston, 1978.
• W. Franz, General Topology, G. Harrap and Co. Ltd. London 1965.
• J. R. Giles, Introduction to the Analysis of Metric Spaces, Cambridge University Press, 1989.
• S.-T. Hu, Introduction to General Topology, Holden-Day Inc. San Francisco, 1966.
• T. Husain, Topology and Maps, Plenum Press, New York, 1977.
• K. D. Joshi, Introduction to General Topology, Wiley Eastern Limited, New Delhi, 1986.
• Ι. Kaplansky, Set Theory and Metric Spaces, Allyn and Bacon Inc., Boston, 1975.
• R. L. Kasriel, Undergraduate Topology, W. B. Saunders Co. Philadelphia, 1971.
• J. L. Kelley, General Topology, D. Van Nostrand Co. Inc., Toronto 1965.
• S. Lipschutz, Theory and Problems of General Topology, Schaum’s Outline Series, New York, 1965.
• Mwndelson, Introduction to Topology, Prentice-Hall Inc. New Jersey, 1975.
• M. G. Murdeshuar, General Topology, Wiley Eastern Limited, New Delhi, 1986.
• M. H. A. Newman, Elements of the Topology of Plane Sets of Points, Cambridge University Press, 1964.
• Α. W. Schurle, Topics in Topology, North Holland, New York, 1979.
• Β. Στάϊκος, Μαθήματα Μαθηματικής Αναλύσεως Μέρος Ι και Μέρος ΙΙ, Ιωάννινα, 1981.

Algebraic Structures I (MAY422)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAY422
Semester	4
Course Title	Algebraic Structures I
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 5, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, English
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 the students to the study algebraic properties of sets which are equipped with one or more (binary) operations. Such mathematical objects are called algebraic structures. We will mainly deal with two types of algebraic structures: Groups. The standard example is the group of permutations of a, usually finite, set. This is the set of all bijective functions from a set to itself endowed with the operation of composition of functions. Rings. The standard example of a ring is the set of integers equipped with the operations of addition and multiplication of integers. We will formulate various theorems concerning the structure and basic properties of groups and rings emphasizing the concept of isomorphism of groups or rings. From the perspective of Algebra two algebraic structures which are isomorphic, they have exactly the same algebraic properties. As a direct consequence, results concerning an algebraic structure are valid in any isomorphic algebraic structure. In the course we present several examples illuminating various notions of symmetry. It should be noted that the notion of symmetry is the central theme which underlies the concept of group/ring. At the end of the course we expect the student: (a) to have understood the definitions and basic theorems which are discussed in the course, (b) to have understood how they are applied in discrete examples, (c) to be able to apply the material in order to extract new elementary conclusions, and finally (d) to perform some (no so obvious) calculations.

General Competences

The course aims to enable the undergraduate student to acquire the ability to analyse and synthesize basic knowledge of the theory of algebraic structures, in particular of the general theory of Groups and Rings, which form an important part of modern algebra. The contact of the undergraduate student with the ideas and concepts of the theory of groups and rings, (a) promotes the creative, analytical and deductive thinking and the ability to work independently, (b) improves his critical thinking and his ability to apply abstract knowledge in various field.

Syllabus

• Preliminaries: Sets, functions, equivalence relations, partitions, (binary) operations.
• Groups – Permutation groups.
• Cyclic groups – generators.
• Cosets with respect to a subgroup – Lagrange’s Theorem.
• Homomorphisms of groups – Quotient groups.
• Rings and fields - Integral domains.
• The theorems of Fermat and Euler.
• Polynomial rings – Homomorphisms of Rings.
• Quotient rings – Prime and maximal ideals.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face to face)
Use of Information and Communications Technology	Teaching Material: Teaching material in electronic form available at the home page of the course. Communication with the students: Office hours for the students (questions and problem solving). Email correspondence Weekly updates of the homepage of the course.
Teaching Methods	Activity Semester Workload Lectures (13x5) 65 Working independently 100 Exercises-Homeworks 22.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students, in English) which includes analysis of theoretical topics and resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Introduction to Statistics (MAY431)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ431
Semester	4
Course Title	Introduction to Statistics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 4, 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, reading Course)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	At the end of the course student should be able to: Understand the meaning of the population and the random sample. Present summary quantitative and qualitative data. Estimate unknown population parameters. Carry out basic statistical hypothesis, and finally, Be able to simply adapt linear regression models and conduct one way analysis of variance.
General Competences	Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism

Syllabus

Descriptive Statistics. Population, Samples & Random Samples. Frequencies, Histograms & Frequencies Statistics. Statistics & Sampling Distributions. χ2, t & F Distributions. Sampling from Normal Populations. Statistical Inference: Parameter Estimation & Tests of Hypotheses. Simple Linear Regression. One-Way & Two-Way Analysis of Variance.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures (13 X 4) 52 Working independently 104 Exercises-Homeworks 31.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English).

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Mendenhall, W., Scheaffer, R. L. and Wackerly, D. D.(1981). Mathematical Statistics with Applications. 2d ed. ISBN: 0-534-98019-8. Duxbury Press. Boston

Introduction to Ordinary Differential Equations (MAY514)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΥ514
Semester	5
Course Title	Introduction to Ordinary Differential Equations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 5, Credits: 7.5)
Course Type	General 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

Bloom's Taxonomy. (1) Remembering: The notion of linear and non-linear ODE. The notion of existence of solutions and uniqueness of solutions for a linear and non-linear ODE. The notion of stability for a linear ODE, for a system of linear ODE's and for a vector linear ODE. (2) Comprehension: Study the existence and uniqueness of solutions of a ODE. Methods for finding the formula of the general solution of an linear ODE and studying their stability. Study systems of linear ODE's. (3) Applying: Study related real world problems. (4) Evaluating: Teaching secondary school courses.

General Competences

Working independently and in groups. Production of free, creative and inductive thinking. Creative, analytic and synthetic thinking.

Syllabus

Section 1. Introduction: Study specific, not necessarily linear, ODE's (Indicatively first order linear, Bernoulli, Riccati), Existence and uniqueness of solutions for first order, not necessarily linear, ODE's (Indicatively Peano Theorem).

Section 2. Study linear ODE's: Methods of calculating formulas of solutions (Method of undetermined coefficients, method of variation of parameters, power series solutions, Laplace transformation), Phase plane, Stability, Transforming a system of ODE's to a vector ODE.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face (Lectures)
Use of Information and Communications Technology	The platform “e-course” of the University of Ioannina
Teaching Methods	Activity Semester Workload Lectures 45 Assignments/Tests 52.5 Individual study 90 Course total 187.5
Student Performance Evaluation	Written Final Examination (Theory and Exercises) 100%

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Χ. Φίλος, Μία Εισαγωγή στις Διαφορικές Εξισώσεις
• R. Agarwal, D. O’Regan, H. Agarwal, Introductory Lectures on Ordinary Differential Equations
• F. Ayres, Differential Equations

Elementary Differential Geometry (MAY522)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAY522
Semester	5
Course Title	Elementary differential geometry
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 5, 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

It is an introductory course on differential geometry. The aim is to introduce and study geometric properties of regular curves (both plane and space) and regular surfaces. Fundamental notions of differential geometry of curves and surfaces are introduced and studied. Among them is the notion of curvature. The study requires tools from Linear Algebra and Calculus of several variables. Upon completion of the course, the student should be familiar with basic notions of differential geometry like the one of curvature, first and second fundamental form, isometries between surfaces and their geometric meaning.

General Competences

Work autonomously Work in teams Develop critical thinking skills

Syllabus

• Plane curves, arclength, curvature, Frenet frame.
• Space curves, curvature and torsion, Frenet frame, fundamental theorem of curves.
• Surfaces, parametrization, Gauss map, Weingarten map, first and second fundamental form, normal curvature, principal and asymptotic directions, Gaussian and mean curvature, minimal surfaces, Theorema Egregium, Gauss and Weingarten formulas, fundamental theorem of surfaces, developable surfaces.

Teaching and Learning Methods - Evaluation

Delivery	Direct
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 65 Autonomous study 127.5 Course total 187.5
Student Performance Evaluation	Written final examination

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Barrett O' Neil, Στοιχειώδης Διαφορική Γεωμετρία, Πανεπιστημιακές Εκδόσεις Κρήτης, 2002
• Manfredo do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976

Complex Functions I (MAY611)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAΥ611
Semester	6
Course Title	Complex Functions I
Independent Teaching Activities	Presentations, exercises, lectures (Weekly Teaching Hours: 5, Credits: 7.5)
Course Type	General 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

It is the most basic introductory course of Mathematical Analysis of the complex space. The student begins to understand the notion of complex numbers and their properties. He/she learns about the use of the complex numbers field in solving some real numbers problems. The student learns about the elementary complex functions and then he/she learns about the line integral as well as the complex integral of such functions. Especially, the advantage of such integrals and their important properties are emphasized. Finally, the student learns the use of complex integrals in computing improper integrals of real functions.

General Competences

Working independently Team work Working in an international environment Working in an interdisciplinary environment Production of new research ideas

Syllabus

The complex plane, Roots, Lines, Topology, Convergence, Riemann sphere, analytic properties of complex functions, Power series, elementary functions (rational, exp, log, trigonometric functions, hyperbolic, functions), line integrals, curves, conformal mappings, homotopic curves, local properties of complex functions, basic theorems, rotation index, General results, singularities, Laurent series, Residuum, Cauchy Theorem, Applications.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	Use of ICT for the presentation and communication for submission of the exercises
Teaching Methods	Activity Semester Workload Lectures 65 Home exercises 22.5 Independent study 100 Course total 187.5
Student Performance Evaluation	Greek. Written exam (100%) on the theory and solving problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Γιαννούλης, Ι. (2024). Μιγαδική Ανάλυση [Προπτυχιακό εγχειρίδιο]. Κάλλιπος, Ανοικτές Ακαδημαϊκές Εκδόσεις. http://dx.doi.org/10.57713/kallipos-408
• R. Remmert. Theory of Complex Functions. Springer, 1998.
• S. Lang. Complex Analysis. Fourth Edition. Springer, 1999.

Classical Mechanics (MAY648)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE648
Semester	6
Course Title	Classical Mechanics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 4, 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 course provides an introduction to theoretical physics, and aims to broaden the knowledge of Mechanics already gained even in secondary education, with the basic criterion being the mathematical formalism of physical problems. Therefore, the course introduces the basic concepts of Classical Mechanics and their application to particles, particle systems and continuous media. Upon completion of this course the students will be able to use advanced mathematics to describe natural phenomena and interpret mathematical results in physical terms. Also, students are expected to develop skills for formulating and solving physical problems.

General Competences

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

Syllabus

Review and connection via physical concepts with the basic tools: areas, mass and density, inertia, center of mass and moments. Review of basic types of differential equations and basic concepts of mechanics (space, time and material point). Newton's axioms and the notion of power. Linear motion, energy and angular momentum. Central forces, many-body systems. Lagrangian and Hamiltonian mechanics.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	Yes
Teaching Methods	Activity Semester Workload Lectures 52 Self study 104 Exercises 31.5 Course total 187.5
Student Performance Evaluation	Final exam

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Κ. Τσίγκανος, Εισαγωγή στη Θεωρητική Μηχανική, Εκδόσεις Σταμούλη, 2004.

Elective Courses

History of Mathematics (MAE501)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE501
Semester	5
Course Title	History of Mathematics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	No
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 Introduction to the History of Mathematics. The course is about the history of Mathematical concepts that are covered in the curriculum of the Elementary school, High school and the first years of the University. There will be also presenations on topics that relate the development of Mathematics with the historical development of other Sciences.
General Competences	Search for, analysis and synthesis of data and information, with the use of the necessary technology Working independently Team work Working in an interdisciplinary environment Production of free, creative and inductive thinking

Syllabus

• Mathematics in Antiquity.
• Mathematics in Ancient Greece.
• Hellenistic Mathematics.
• Mathematics from 150 BC to the Renaissance in different civilizations.
• Topics on the History of Contemporary Mathematics.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Communication with students Use of ICT in teaching
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Working independently 38 Exercises-Homeworks 73 Course total 150
Student Performance Evaluation	Language of evaluation: Greek Written Examination, Oral Presentation, written assignments in Greek

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Teaching of Mathematics (MAE503) (also MAE602)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE503 (also MAE602)
Semester	5 (also 6)
Course Title	Teaching of Mathematics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	-
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	-
General Competences	Working independently Team work Project planning and management Respect for difference and multiculturalism Showing social, professional and ethical responsibility and sensitivity to gender issues Criticism and self-criticism Production of free, creative and inductive thinking

Syllabus

Education and its scopes. Elements of the History of Mathematics and evolution of Mathematics Education. Genaral scopes of Mathematics Education. Philosophy and learning of Mathematics. Models and methods in the teaching of Mathematics. The teaching of mathematical notions: analysis, algebra, geometry. Planning, preparation, presentation, evaluation. Teaching outcomes: conclusions and aspects. The structure of Mathematics Education in the secondary level. Mathematical journals and competitions.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Study of literature 36 Presentations 25 Individual work 25 Team work 25 Course total 150
Student Performance Evaluation	Public presentation, written work, essay/report, semester examination.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ΜΠΑΜΠΗΣ ΤΟΥΜΑΣΗΣ, Σύγχρονη Διδακτική των Μαθηματικών, Gutenberg, Αθήνα1999
• ΑΘΑΝΑΣΙΟΣ ΓΑΓΑΤΣΗΣ, Θέματα Διδακτικής των Μαθηματικών, Εκδόσεις Κυριακίδη, Θεσσαλονίκη 1993
• MORRIS KLEIN, Γιατί δεν μπορεί να κάνει πρόσθεση ο Γιάννης, Εκδόσεις Βάνιας, Θεσσαλονίκη 1993
• G. POLYA, How to solve it?, Princeton University Press, 1999
• PIERRE VAN HIELE, Structure and Insight: A Theory of Mathematics Education, Academic Press, 1986
• SUE JOHNSTON-WILDER, PETER JOHNSTON-WILDER, DAVID PIMM, CLARE LEE, Learning to Teach Mathematics in the Secondary School, 3 rd Edition, Routledge, 2011 (also MAE602)

Elements of General Topology (MAE513)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE513
Semester	5
Course Title	Elements of General Topology
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 to introduce the student to basic notions of General Topology and, in some way, to generalize already obtained knowledge on metric spaces. It is an optional course for students interested in having a background on pure mathematics. It is also attempted to broaden students horizon to mathematical structures which, even if they seem abstract, they have important applications in several branches of science.

General Competences

Analysis and synthesis of data and information Working independently Team work Working in an interdisciplinary environment Production of free, creative and inductive thinking Production of new research ideas

Syllabus

The notion of Topology. Topologies from metrics and non-metrizable topologies. Bases and subbases. Fundamental notions (open sets, closed sets, closure, interior, boundary, accumulation points). Neighborhood bases and systems. Convergence of sequences in topological spaces. Nets and convergence of nets. Continuity. Topologies from sequence of functions, product spaces. Spaces of 1 and 2 countability. Separation (T1, T2, T3, T4 spaces). Compactness of topological spaces.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	Use of special software (tex, mathematica, e.t.c.) for presentation of projects and exercises.
Teaching Methods	Activity Semester Workload Lectures (6x3) 18 Seminars (7x3) 21 Individual study 78 Exercises/projects 33 Course total 150
Student Performance Evaluation	Greek or English Public presentation Final written exam Criteria for evaluation are posted on course's site (E-course) at the beginning of each semester.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• J. L. Kelley, General Topology, D. Van Nostrand Co. Inc., Toronto 1965
• J. Dugudji, Topology, Allyn and Bacon Inc., Boston 1978
• K. D. Joshi, Introduction to General Topology, Wiley Eastern Limited, New Delhi, 1986

Topics in Functions of One Variable (MAE515)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ515
Semester	5
Course Title	Topics in Functions of One Variable
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 undergraduate student of special theoretical background in the theory of real functions.
General Competences	The objective of the course is the undergraduate student's ability achievement in analysis and synthesis of the basic background in the theory of real functions.

Syllabus

Monotone functions - points of continuity, functions of bounded variation, sets of measure zero, Lebesgue's theorem (every monotone function is differentiable almost everywhere), Darboux continuous functions-definitions, properties, equivalent characterizations, convex functions, semicontinuous functions, continuity points of Riemann integrable functions, Baire classes, Borel measurable functions, analytic sets-characterizations, connections with Borel sets-related theory, Lebesgue integral, Stieltjes integral.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 78 Exercises solutions 33 Course total 150
Student Performance Evaluation	Written examination at the end of the semester.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• A.C.M. Van Rooij, W.H. Schikhof, Α second course on real functions, Cambridge University Press.

Algebraic Curves (MAE521)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE521
Semester	5
Course Title	Algebraic Curves
Independent Teaching Activities	Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)
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 students will acquire with the successful completion of the course the basic theory of Algebraic curves and the ability to solve problems on Algebraic curves.
General Competences	The course aim is for the student to acquire the ability in analysis and synthesis of knowledge in algebraic curves and produces free, creative and inductive thinking.

Syllabus

Affine plane, polynomial rings, unique Factorization Domains, resultants, Rational curves and Applications, Projective space, tangents, singular points, asymptotes. Intersection multiplicity, Bezout's Theorem, Linear Systems. Pascal's Theorem. Nine points Theorem. Inflection points. Elliptic Curves.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which includes resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Group Theory (MAE525)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE525
Semester	5
Course Title	Group Theory
Independent Teaching Activities	Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special background, skills development.
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, 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

Familiarity with: group, abelian group, subgroup, normal subgroup, quotient group, direct product of groups, homomorphism, isomorphism, kernel of a homomorphism.  Apply group theory to describe symmetry, describe the elements of symmetry group of the regular n-gon (the dihedral group D2n). Compute with the symmetric group. Know how to show that a subset of a group is a subgroup or a normal subgroup. State and apply Lagrange's theorem. State and prove the isomorphism theorems. Sylow theorems. The classification of finite abelian groups. Normal series, central series, nilpotent groups. Applications in Geometry.

General Competences

Study particular characteristics of group theory in topology and geometry. Independent and team work. Working in an interdisciplinary.

Syllabus

• Basic properties in groups.
• Symmetries.
• Subgroups, Direct products, Cosets.
• Symmetric groups.
• Normal Subgroups, Quotient groups.
• Homomorphisms.
• Semidirect product.
• Classification of finite abelian groups.
• Sylow theorems.
• Normal series, Solvable groups. Central series, Nilpotent groups.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Communication with students
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Written Examination, Oral Presentation, written assignments in Greek (in case of Erasmus students in English) which includes resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• An Introduction to the Theory of Groups (Graduate Texts in Mathematics) 4th Edition by Joseph Rotman.

Groebner Bases (MAE526)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE526
Semester	5
Course Title	Groebner Bases
Independent Teaching Activities	Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)
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 students will acquire with the successful completion of the course the skills to apply polynomial division the skills to compute Groebner bases the skills to apply Groebner bases techniques to problems coming from elimination theory, Algebraic Geometry, filed extensions, Graph Theory and Integer programming.
General Competences	The course aim is for the student to acquire the ability in analysis and synthesis of knowledge in Computational Algebra and produces free, creative and inductive thinking.

Syllabus

Polynomial rings. Hilbert;s basis Theorem. Noetherian rings. Monomial οrders. Division Alghorithm. Groebner bases. S-polynomials and Buchberger;s alghorithm. Irreducible and universal Groebner bases. Nullstellensatz Theorem. Applications of Groebner: bases in elimination, Algebraic Geometry, field extensions, Graph Theory and Integer Programming.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which includes resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Geometry of Transformations (MAE527)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE527
Semester	5
Course Title	Geometry of Transformations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	Linear Algebra, Analytic Geometry
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 can be viewed as a continuation of Analytic Geometry. The aim is to study geometric transformations of the plane or space. The classification of isometries is provided. Further applications are given, as well the classification of second degree surfaces. Moreover, algebraic curves are studied. Upon completion of the course, the student should be familiar with notions of geometry and geometric transformations that are used in other courses like Calculus of several variables.

General Competences

Work autonomously Work in teams Develop critical thinking skills.

Syllabus

Geometric transformations of the plane and space. Isometries, applications. Classification of second degree surfaces. Algebraic curves.

Teaching and Learning Methods - Evaluation

Delivery	Direct
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Autonomous study 111 Course total 150
Student Performance Evaluation	Written final examination.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Thomas F. Banchoff και John Wermer, Η Γραμμική Άλγεβρα μέσω Γεωμετρίας, Εκδόσεις Leader Books, Σειρά Πανεπιστημιακά Μαθηματικά Κείμενα, Αθήνα, 2009

Theory of Probability and Statistics (MAE531)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ531
Semester	5
Course Title	Theory of Probability and Statistics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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

Extension and generalization of concepts taught in MAF331 and MAF43# Creation of a suitable base for deepening the scope of Statistical Science. At the end of the course the student should be able to: Model procedures and situations that occur in everyday reality or in other scientific areas in the Theory of Probability. Understand the basic limit theorems of Probability Theory (laws of large numbers, central limit theorem) and use them for approximating probability calculations. Find the distribution of a function of random variables. Make basic calculations of probability, averages, dispersions, etc., in problems involving randomness with more than one random variable.

General Competences

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

Syllabus

Random vectors-Multivariate distribution function-Joint probability- Joint probability density function. Marginal distributions. Conditional distributions. Special bivariate and multivariate distributions (multinomial, bivariate and multivariate normal etc). Expectation, Variance-Covariance matrix. Moments and Moment generating function of random vector. Distribution of a function of random variables. Order Statistics. Convergence of random variables. Sampling distributions.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English).

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Mood, A. M., Graybill, F. A. and Boes, D. C. (1974). Introduction to the Theory of Statistics. 3d ed. ISBN-13 978007085465# McGraw-Hill. New York.

Stochastic Processes (MAE532)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ532
Semester	5
Course Title	Stochastic Processes
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 term "stochastic" is used to describe phenomena in which some randomness inherent. A stochastic process is a probabilistic model that describes the behaviour of a system that randomly evolves over time. Observing the system at discrete points in time (for instance at the end of each day or at the end of a time period, etc.) one gets a discrete time stochastic process. Observing the system continuously through time one gets a continuous time stochastic process. Objectives of the course are: Understanding the behaviour of a real system and based on its study to derive reliable results, a careful analysis of the model and the calculation of the results. A variety of classes of stochastic processes such as, the random walk, the Markov chains etc is used. The student should be able to understand the meaning of the stochastic process, use the Markov processes for modelling systems and become familiar with their application, and be able to make various calculations and appropriate conclusions when the stochastic process describes a specific applied problem.

General Competences

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

Syllabus

Random Walk: Simple random walk, absorbing barriers, reflecting barriers. Markov Chains: General definitions, classification of states, limit theorems, irreducible chains. Markov Processes: The birth-death process. 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-Homeworks 33 Course total 150
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which concentrates on the solution of problems which are motivated by the main themes of the course.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• R. Dobrow. Introduction to Stochastic Processes with R, Wiley, 2016.
• R. Durret. Essentials of Stochastic Processes, Springer, 3rd edition, 2016.
• V.G. Kulkarni. Modeling and Analysis of Stochastic Systems, 3rd edition, CRC Press, London 2017.
• N. Privault. Understanding Markov Chains [electronic resource] HEAL-Link Springer ebooks, 2013 (Κωδικός Εύδοξου: 73260010).
• M. Pinksy, S. Karlin. An introduction to stochastic modelling, 4th edition, Academic Press, 2011.
• S. Ross. Introduction to probability models, Academic Press, New York, 2014.
• [Περιοδικό / Journal] Stochastic Processes and their Applications (Elsevier)
• [Περιοδικό / Journal] Stochastics (Taylor - Francis)
• [Περιοδικό / Journal] Journal of Applied Probability (Cambridge University Press)

Introduction to Computational Complexity (MAE542)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE542
Semester	5
Course Title	Introduction to Computational Complexity
Independent Teaching Activities	Lectures, exercises, tutorials (Weekly Teaching Hours: 3, Credits: 6)
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 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-Homeworks 33 Course total 150
Student Performance Evaluation	Final written examination (70%) Exercises / Homework (30%)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• 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.

Applied Tensor Analysis (MAE543)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ543
Semester	5
Course Title	Applied Tensor Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 concepts of Tensor Analysis. The objectives of the course are: Development of the theoretical background in matters relating to Tensor Analysis. Ability of the student to apply the basic concepts of Tensor Analysis. Upon completion of this course the student will be able to solve with analytical methods simple problems of Tensor Analysis and deepen further understanding of such methods.

General Competences

The course aims to enable the undergraduate students to develop basic knowledge of Applied Tensor Analysis and in general of Applied Mathematics. The student will be able to cope with problems of Applied Mathematics giving the opportunity to work in an international multidisciplinary environment.

Syllabus

The tensor concept, Invariance of tensor equations, Curvilinear coordinates, Tensors in generalized curvilinear coordinates, Gauss, Green and Stokes theorems, Scalar and vector fields, Nabla operator and differential operators, Covariant differentiation, Integral theorems, Applications to Fluid Dynamics.

Teaching and Learning Methods - Evaluation

Delivery	In class
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Study of theory 78 Home exercises 33 Course total 150
Student Performance Evaluation	Weekly assignments Final project Written examination at the end of the semester

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• A. I. Borisenko and I. E. Taparov, Vector and Tensor Analysis, Edition: 2/2017, Editor: G. C. FOYNTAS (in Greek).
• H. Lass, Vector and Tensor Analysis, Edition: 2/2017, Editor: G. C. FOYNTAS (in Greek).

Logic Programming (MAE544)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE544
Semester	5
Course Title	Logic Programming
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 goal of this course is the deeper understanding of PROLOG. During the course a detailed examination of the following topics are done: Procedural and Declarative Programming Logic Programming a version of Declarative Programming The programming language PROLOG (PROLOG programs syntax, Lists, Operators, Arithmetic, Backtracking control, The negation in PROLOG, Recursive predicates, Data Structure manipulation, PROLOG implementation to searching problems, symbolic processing, natural language understanding and metaprogramming) Logic Programming Theory Logic Programming under restrictions Logic Programming systems implementation techniques Parallel Logic Programming Logic Programming for knowledge representation After completing the course the student can handle: programming in PROLOG solving exercises in PROLOG tracking applications in PROLOG

General Competences

Programming in PROLOG Implement PROLOG to Mathematics, Natural Language, Expert Systems, e.t.c. Implementation- Consolidation.

Syllabus

• Introductory concepts of Automata , Computability and Complexity as well as basic definitions, basic theorems and inductive proofs
• Finite State Machines and Languages, Finite Automata (Deterministic FA, Nondeterministic FA, FA with Epsilon-Transitions) and their applications, Regular Expressions and Languages, derivation trees. Removing Nondeterminism . Equivalence NFA and NFA with ε-moves. Minimization of DFA, Pumping Lemma
• FA and Grammars. Grammars of Chomsky Hierarchy. Regular Sets (RS). Properties of Regular Languages. RS and FA. Finding a correspondence Regular Expression of a FA. Abilities and disabilities of FA.
• Context-Free Grammars and Languages, Pushdown Automata (Deterministic PDA, Acceptance by Final State, Acceptance by Empty Stack) , Properties of Context-Free Languages. Correspondence PDA and Context-Free Languages.
• Introduction of Turing Machines. Standard TM, useful techniques for TM constructions. Modification of TM. TM as procedure.
• Unsolvability. The Church-Turing Thesis. The Universal TM. The Halting Problem for TM. Computational Complexity. NP-complete problems.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	Yes, Use of Natural Language and Mathematical Problems Processing Laboratory
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Exercises 33 Course total 150
Student Performance Evaluation	Final test

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Π. Σταματόπουλος, "Λογικός και Συναρτησιακός Προγραμματισμός", Σύνδεσμος Ελληνικών Ακαδημαϊκών Βιβλιοθηκών, http://hdl.handle.net/11419/3587 (με διορθωμένα παροράματα εδώ)
• Η. Σακελλαρίου, Ν. Βασιλειάδης, Π. Κεφαλάς, Δ. Σταμάτης, "Τεχνικές Λογικού Προγραμματισμού", Σύνδεσμος Ελληνικών Ακαδημαϊκών Βιβλιοθηκών, http://hdl.handle.net/11419/777
• I. Bratko, "Prolog Programming for Artificial Intelligence", Third Edition, Addison-Wesley, 2000.
• L. Sterling, E. Shapiro, "The Art of Prolog", The MIT Press, 1994.
• J. W. Lloyd, "Foundations of Logic Programming", Springer Verlag, 1993

Biomathematics (MAE546A)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ546A
Semester	5
Course Title	Biomathematics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 is an introduction to the basic concepts of Biomathematics. Upon successful completion of the course, the student will be able to: apply basic concepts of biomathematics understand and apply advanced analytical and approximate techniques to biomathematics problems critically analyze and compare the effectiveness of methods and deepen their further understanding combine advanced techniques to solve new problems in biomathematics

General Competences

The course aims to enable the student to analyze and synthesize basic knowledge of Biomathematics and Applied Mathematics. Search for, analysis and synthesis of data and information, with the use of the necessary technology Adaptation to new situations Autonomous work Decision making Work in an interdisciplinary environment

Syllabus

• Short introduction of Algebra, Analysis and Differential Equations
• Differential equations of biofluids motion
• Applications of mathematical modeling of biofluids in the human body and in the arterial system
• Analytical and numerical techniques for solving the differential equations describing biofluids flows
• Algbraic statistics for Computational Biology: Algebraic varieties and Groebner bases, Toric ideals and varieties, Linear and toric models
• Markov bases, Markov bases for hierarchical models, Contigency tables, Phylogenetic Models.

Teaching and Learning Methods - Evaluation

Delivery	In class
Use of Information and Communications Technology	Provision of study material through the ecourse Communication with students through e-mails, and the ecourse and MS Teams platforms
Teaching Methods	Activity Semester Workload Lectures 39 Study of theory 78 Home exercises 33 Course total 150
Student Performance Evaluation	Weekly assignments Final project Written examination at the end of the semester

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Algebraic Statistics for Computational Biology, L. Pachter, B. Sturmfels, 2005, Editor: Cambridge University Press
• Cardiovascular Mathematics, Modeling and simulation of the circulatory system, Formaggia L., Quarteroni A., Veneziani A., 2009, Editor: Springer

Design and Analysis of Algorithms (MAE581)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE581
Semester	5
Course Title	Design and Analysis of Algorithms
Independent Teaching Activities	Lectures, laboratory exercises, tutorials, quiz (Weekly Teaching Hours: 3, Credits: 6)
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 aims at introducing to students the philosophy of fundamental algorithmic background and techniques. After successfully passing this course the students will be able to: Understand basic algorithmic techniques Analyze complex algorithms Design new algorithmic tools Combine already-known techniques for solving new algorithmic problems
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

• Fundamental concepts of design and analysis of algorithms
• Analysis of algorithms, Asymptotical growing functions
• Typical running times and data structures (lists, arrays, queues, stacks)
• Stable matching, correctness, priority queue
• «Divide & Conquer» technique, sorting, recursive formulations
• Graph algorithms: BFS, DFS, connectedness, topological ordering
• Greedy algorithms: interval scheduling & shortest paths (Dijkstra)
• Minimum spanning trees(Prim & Kruskal algorithms), Huffman coding
• Dynamic programming: maximum flow, interval scheduling, and Knapsack
• Further Topics: computational complexity and ΝΡ-completeness.

Teaching and Learning Methods - Evaluation

Delivery	Lectures
Use of Information and Communications Technology	Use of projector and interactive board during lectures. Course website maintenance. Announcements and posting of teaching material (lecture slides and notes, programs). Announcement of assessment marks via the ecourse platform by UOI.
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Team work 33 Course total 150
Student Performance Evaluation	Final written examination (70%) Design and analyze algorithms Exercises (30%) Design and analyze algorithms

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Approximation Theory (MAE585)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ585
Semester	5
Course Title	Approximation Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 the basic theory of approximation in spaces of functions, be aware and apply the taught methods for best uniform polynomial approximation, least squares polynomial approximation of functions defined in an interval (continues case), as well as of functions defined in a set of points (discrete case), be aware and apply the taught methods for cubic splines polynomial interpolation, 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

Introduction to Approximation Theory in Spaces of Functions (Existence - Uniqueness). Polynomial Approximation of Functions: Weierstrass Theorem. Best Uniform Approximation. Least Squares Approximation. Hermite Polynomial Interpolation. Cubic Splines Polynomial Interpolation.

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 bibliografy 104 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Written examination

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• "Approximation Theory". Noutsos D., University of Ioannina.

Integral Equations (MAE613)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE613
Semester	6
Course Title	Integral Equations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 an introduction to the area of Integral Equations. Students are expected to obtain basic knowledge on standard types of integral equations, learn how to solve certain linear integral equations, also study existence and uniqueness of solutions by the use of fixed point theorems.
General Competences	Working independently Team work Production of free, creative and inductive thinking Production of analytic and synthetic thinking

Syllabus

An introduction with historical notes. Classification of Integral Equations. Problems leading to integral equations. Laplace transformations and their use to solving integral equations. Other integral transformations. Volterra integral equations: Neumann series, successive approximations, Laplace transformation and the convolution kernel. Fredholm integral equations: Symmetric kernels, separated kernels, Fredholm Alternative, classical Fredholm theory. Green functions for second order boundary value problems. Existence and uniqueness of solutions: Banach spaces, contractions and applications to integral equations. Existence of solutions by Schauder's theorem.

Teaching and Learning Methods - Evaluation

Delivery	Lectures. Presentations in class.
Use of Information and Communications Technology	Use of the platform “E-course” of the University of Ioannina
Teaching Methods	Activity Semester Workload Lectures/Presentations 39 Assignments 33 Individual study 78 Course total 150
Student Performance Evaluation	Students choose evaluation by one or both of the following: Class presentation - Essays - Assingments 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 are accessible to students through the platform “E-course” of the University of Ioannina.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Σ. Ντούγια, Ολοκληρωτικές Εξισώσεις
• C. Corduneanu, Principles of Differential and Integral Equations

Ordinary Differential Equations I (MAE614)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE614
Semester	6
Course Title	Differential Equations I
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special 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

Bloom's Taxonomy. (1) Remembering: The notion of fixed point, of maximum domain of solutions and of the stability of not necessarily linear ODE's. (2) Comprehension: Study fixed point theorems and Topological Degree Theory, with applications to not necessarily linear ODE's. Study the maximum domain of solutions of not necessarily linear ODE's. Linearization of ODE's. (3) Applying: Study related real world problems. (4) Evaluating: Teaching secondary school courses.

General Competences

Working independently and in groups. Production of free, creative and inductive thinking. Creative, analytic and synthetic thinking.

Syllabus

Study of not necessarily linear ODE's: Existence of solutions using fixed point theorems and topological degree theory (i.e. Brouwer degree), Maximum domain for solutions, Stability using the Lyapunov method, Linearization.

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, 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 (7x3) 21 Seminars (6x3) 18 Individual study 78 Exrecises/projects 33 Course total 150
Student Performance Evaluation	Language of evaluation: Greek and English. Methods of evaluation: Weekly written assignments. Few number of tests during the semester. Based on their grades in the aforementioned weekly assignments and tests, limited number of students can participate in exams towards the end of the semester, before the beginning of the exams period. In any case, all students can participate in written exams at the end of the semester, during the exams period. 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

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Topics in Real Analysis (MAE615)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE615
Semester	6
Course Title	Topics in Real Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 plan of the course is the achievement by the undergraduate student of the introductory background in the theory of metric spaces.
General Competences	The objective of the course is the undergraduate student's ability achievement in analysis and synthesis of the basic background in Real Analysis.

Syllabus

Baire spaces, the theorem of Cantor, characterization of complete metric spaces, compact metric spaces, Lebesgue's lemma, uniform continuous functions and extensions of them, completetion of a metric space and uniqueness up to isometry, oscillation of a function, continuity sets of a function which is the pointwise limit of a sequence of continuous functions, uniform convergence of a sequence of functions and related topics, Dini's theorem.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 78 Exercises solutions 33 Course total 150
Student Performance Evaluation	Written examination at the end of the semester.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Charalambos D. Aliprantis, Owen Burkinshaw, Principles of Real Analysis, Academic Press.

Measure Theory (MAE616)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE616
Semester	6
Course Title	Measure Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special background
Prerequisite Courses	None (from the typical point of view). In order to be able to follow this course, the knowledge from the following courses are required: Infinetisimal Calculus I, Introduction to Topology.
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (exams in English are provided for foreign students)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

After completing this course the students will Have knowledge of the basic properties of σ-algebras, of measures and especially of Lebesgue measure on the set R of real number and on the Euclidean space R^k. Know the basic properties of measurable functions, the definition of Lebesgue integral in a random measure space. Be able to apply the basic theorems concerning Lebesgue intergral (Monotone Convergernce Theorem, Dominated Convergence Theorem). Understand the difference between Riemann integral and Lebesgue integral on R.

General Competences

The course promotes inductive and creative thinking and aims to provide the student with the theoretical background and skills to use measure theory and integration.

Syllabus

Algebras, σ-algebras, measures, outer measures, Caratheodory's Theorem (concerning the construction of a measure from an outer measure). Lebesgue measure, definition and properties. Measurable functions. Lebesgue integral, Lebesgue's Monotone Convergernce Theorem, Lebesgue's Dominated Convergence Theorem. Comparison between Riemann integral and Lebesgue integral for functions defined on closed bounded integrals of the set of reals.

Teaching and Learning Methods - Evaluation

Delivery	Teaching on the blackboard by the teacher.
Use of Information and Communications Technology	Communication with the teacher by electronic means (i.e. e-mail).
Teaching Methods	Activity Semester Workload Lectures 39 Personal study 78 Solving exercises 33 Course total 150
Student Performance Evaluation	Exams in the end of the semester (mandatory), potential intermediate exams (optional), assignments of exercises during the semester (optional).

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Measure Theory, Donald Cohn, Birkhauser.

Real Analysis (MAE617)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE617
Semester	6
Course Title	Real Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 course aims in presenting topics concerning real valued functions defined on a metric space. Pointwise and uniform convergence of a sequence of functions are discussed as so as topics like Ascoli-Arzela theorem and Stone-Weirstrass theorem. Applications of the above are also given.
General Competences	Working independently Team work Working in an international environment Working in an interdisciplinary environment Production of new research ideas.

Syllabus

Function spaces on a metric space (X,d), pointwise and uniform convergence of sequence of functions, the space B(X) of real bounded functions on X-, the space C(X) of continuous functions on X – equicontinuous subsets of C(X), Ascoli-Arzela theorem and applications, Dini's theorem, Stone-Weierstrass theorem and applications, separable metric spaces, Lindelof's theorem on Euclidean spaces, the Cantor set, the Cantor function-applications.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	Use of ICT for the presentation and communication for submission of the exercises
Teaching Methods	Activity Semester Workload Lectures 39 Home exercises 30 Independent study 81 Course total 150
Student Performance Evaluation	Written examination at the end of the semester.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

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

Seminar in Analysis I (MAE618)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ618
Semester	6
Course Title	Seminar in Analysis I
Independent Teaching Activities	Lectures (Weekly Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	None. However it is desirable to have a strong knowledge of basic notions of differential equations.
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	No
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The ability to write a complete report on a scientific subject. The ability to present this report. Working in teams. The report can be, but not required to be, original. Further details can be determined by the teaching professor.
General Competences	Αnalyse and combine data and information using various technologies. Working independently and in groups. Free, creative, analytic, and conclusive thinking. Decision making. Further details can be determined by the teaching professor.

Syllabus

In depth study in a scientific subject related to mathematical analysis. Further details can be determined by the teaching professor.

Teaching and Learning Methods - Evaluation

Delivery	Details will be determined by the teaching professor. Methods include presentations contacted by the students.
Use of Information and Communications Technology	Details will be determined by the teaching professor.
Teaching Methods	Activity Semester Workload Study in class 39 Other activities determined by the teaching professor 111 Course total 150
Student Performance Evaluation	There is no final exam. Each student must write a report on a specific subject. Each student must present the report publically. Students may miss up to 3 lectures. Other means of evaluation can be determined by the teaching professor.

Attached Bibliography

Bibliography is suggested by the teaching professor, depending on the subject under study.

Differentiable Manifolds (MAE622)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE622
Semester	6
Course Title	Differentiable Manifolds
Independent Teaching Activities	Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, 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

In this lecture, the fundamental concept of a differentiable manifold will be developed. In particular, this lecture is a basic prerequisite for the upcoming class of Riemannian geometry. After a quick review of basic facts from general topology we will introduce the notions of a smooth manifold, tangent bundle, vector field, submanifold, connection, geodesic curve, parallel transport and Riemannian metric. On the completion of this course we expect that the students fully understand these important concepts and the main theorems that will be presented in the lectures.

General Competences

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

Syllabus

Review of basic facts from general topology, smooth manifolds, tangent bundle, vector fields, immersions and embeddings, Lie bracket, Frobenius' theorem, Whitney's embedding theorem, connections and parallel transport, Riemannian metrics.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Weekly exercises and homeworks, presentations, final written exam in Greek (in case of Erasmus students in English) which includes resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• M. do Carmo, Riemannian Geometry, Birkhaüser Boston, Inc., Boston, MA, 1992.
• V. Guillemin & A. Pollack, Differentiable Topology, Prentice-Hall, Inc, Englewood Cliffs, 1974.
• J. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218, 2013.
• J. Milnor, Topology From the Differentiable Viewpoint, Princeton University Press, NJ, 1997.
• L. Tu, An Introduction to Manifolds, Universitext. Springer, New York, 2011.
• Δ. Κουτρουφιώτης, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων, 1994.

Elementary Global Differential Geometry (MAE624)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE624
Semester	6
Course Title	Elementary Global Differential Geometry
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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

It is an introductory course on global differential geometry. The aim is to study global geometric properties of regular plane curves and regular surfaces. The study requires tools from Linear Algebra, Calculus of several variables, Topology and elementary differential geometry. On completion of the course the student should be familiar with the interplay between local and global properties of curves and surfaces.

General Competences

Work autonomously Work in teams Develop critical thinking skills

Syllabus

Convex curves, Hopf's Umlaufsatz, Four vertex theorem, isoperimetric inequality. Surfaces, vector fields, covariant derivative, parallel transport, geodesic curvature, geodesics, exponential map, surfaces of constant Gaussian curvature, Gauss Bonnet Theorem, Liebmann Theorem.

Teaching and Learning Methods - Evaluation

Delivery	Direct
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Autonomous study 111 Course total 150
Student Performance Evaluation	Written final examination

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Barrett O' Neil, Στοιχειώδης Διαφορική Γεωμετρία, Πανεπιστημιακές Εκδόσεις Κρήτης, 2002
• Manfredo do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976

Rings, Modules and Applications (MAE628)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE628
Semester	6
Course Title	Rings, Modules and Applications
Independent Teaching Activities	Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)
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 principal aim of the course is to introduce the students to the main tools and methods of the theory of modules and rings. At the end of the course we expect the student to have understood the definitions and basic theorems which are discussed in the course, to have understood how they are applied in discrete examples, to be able to apply the material in order to extract new elementary conclusions, and finally to perform some (no so obvious) calculations.

General Competences

The contact of the undergraduate student with the ideas and concepts of the theory of modules and rings, (a) promotes the creative, analytical and deductive thinking and the ability to work independently, (b) improves his critical thinking and his ability to apply abstract knowledge in various field.

Syllabus

• Elementary Ring Theory.
• Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains.
• Module Theory.
• Modules over polynomial rings.
• Finitely generated and free modules.
• Modules over Principal Ideal Domains.
• Decomposition Theorems.
• Applications to Linear Algebra and Abelian groups.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which includes resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Μ. Μαλιάκας: «Εισαγωγή στη Μεταθετική Άλγεβρα»,  Εκδόσεις Σοφία. 
• N. Jacobson: “Basic Algebra I”, Dover Publications (1985).
• S. Lang: «Άλγεβρα», Εκδόσεις Πολιτεία (2010).

Linear Programming (MAE631K)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ631K
Semester	6
Course Title	Linear Programming
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 course learning outcomes are: the introduction of the students to linear programming formulation, the comprehension of the mathematical properties of linear programming problems, the understanding of the theory underlying the simplex algorithm, the understanding of the dual theory and its interpretation, the use of LINDO software package to solve linear programming problems Upon successful completion of the course the student will be able to: to model linear programming problems. to solve linear programming problems with the Simplex method. to apply the appropriate modifications of Simplex method when it is necessary. to validate and interpret the results obtained when linear programming problems are solved using LINDO software

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

• Formulating linear programming problems
• Graphical solution
• The Simplex Method 
• The bib M method
• The Two-Phase Simplex Method
• Dual theory
• Sensitivity analysis
• Transportation problem
• Assignment problem
• Transshipment problem
• Other network problems

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	Lindo Software, Email, class web, MSTeams
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 78 Fieldwork (3-4 set of homework) 33 Course total 150
Student Performance Evaluation	LANGUAGE OF EVALUATION: Greek METHODS OF EVALUATION: Final exam (100%)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Βασιλείου Β. και Τσαντας Ν., Εισαγωγή Στην Επιχειρησιακή Έρευνα, Εκδόσεις Ζητη 2000.
• Κολετσος Κ., Στογιαννης Δ. Επιχειρησιακή Έρευνα Θεωρία, Αλγόριθμοι Και Εφαρμογές, Εκδόσεις, Συμεών, 2021.
• Κουνιας Σ. και Φακινος Φ., Γραμμικός Προγραμματισμός, Εκδόσεις Ζητη, Θεσσαλονίκη 1999.
• Παπαρριζος Π., Γραμμικός Προγραμματισμός. Εκδόσεις Ζυγός, Θεσσαλονίκη 1999.
• Σισκος Γ., Γραμμικός Προγραμματισμός, Εκδόσεις Νέων Τεχνολογιών, Αθήνα 1998.
• Υψηλαντης Π. Μέθοδοι και Τεχνικές Λήψης Αποφάσεων, Εκδόσεις Προπομπός, 2015.
• Φακινου Δ. Και Οικονομου Α., Εισαγωγή Στην Επιχειρησιακή Έρευνα- Θεωρία Και Ασκήσεις, Αθήνα 2003.
• Βertsimas D. And J. N. Τsitsiklis Introduction to Linear Optimization, Athena Scientific, 1997.
• Κουνετάς, Κ., Χατζησταμούλου, Ν., 2015. Εισαγωγή Στην Επιχειρησιακή Έρευνα Και Στον Γραμμικό Προγραμματισμό. Λύσεις Προβλημάτων Με Το Πρόγραμμα R. [Ηλεκτρ. Βιβλ.] Αθήνα: Σύνδεσμος Ελληνικών Ακαδημαϊκών Βιβλιοθηκών. Διαθέσιμο Στο: Http://Hdl.Handle.Net/11419/5699.
• Gass S. Linear Programming Methods and Applications, Mcgraw-Hill 1985.
• Hadley G. Linear Programming, Addison-Wesley Publishing Company, Inc, 1965.
• Hillier F. S. And G. J. Lieberman Introduction Operations Research. The Mcgraw-Hill Companies, 2001.
• RardinL. R. Βελτιστοποίηση στην Επιχειρησιακή Έρευνα, ΕκδοσειςΚλειδαριθμος, 2022.
• TahaH., Εισαγωγή Στην Επιχειρησιακή Έρευνα, 10η Έκδοση, Eκδόσεις Α. Τζιολα & YιοιA.E., 2017.
• Winston W. L., Operations Research (Applications And Algorithms). Duxbury Press (International Thomson Publishing) 1994.
• [Περιοδικό / Journal] Mathematical Programming Journal, Series A and Series B.
• [Περιοδικό / Journal] INFORMS Transactions on Education (ITE).

Statistical Inference (MAE633)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ633
Semester	6
Course Title	Statistical Inference
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 aim of the course is to present and study techniques and methods of parametric statistical inference. In particular, the interest is mainly focused on the theoretical development of the field of parameter estimation (point and interval) and the development of the theory of statistical tests for testing statistical hypotheses. Moreover, this course aims to provide the necessary tools and methods which help students to be able to draw statistical conclusions on the basis of experimental data and by utilizing these methods. At the end of the course students will have acquired the theoretical background of the parametric statistical inference methodologies.

General Competences

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

Syllabus

Point estimation: unbiased, sufficient and efficient estimators, unbiased estimators with minimum variance, the Cramer-Rao lower bound for the variance, Lehmann-Scheffe theory, asymptotic properties of estimators, methods of estimation (method of maximum likelihood and method of moments). Interval estimation. Confidence intervals. Testing Statistical Hypothesis: the Neyman- Pearson lemma, simple and composite hypotheses, uniformly most powerful tests, likelihood ratio tests. Large sample tests. 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-Homeworks 33 Course total 150
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which concentrates on the solution of problems which are motivated by the main themes of the course.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Casella, G. and Berger, R. (2002). Statistical Inference. 2 Edition. Duxbury Advanced Series.
• Hogg, R. V., McKean, J. W. and Craig, A. T. (2005). Introduction to Mathematical Statistics. Pearson Education, Inc.
• Mood, A., Graybill, F. and Boes, D. (1974). Introduction to the Theory of Statistics. McGrawHill.
• Roussas, G. (2003). An Introduction to Probability and Statistical Inference. Academic Press.
• Κουρούκλης, Σ. (2007). Στατιστική Ι. Πανεπιστήμιο Πατρών.

Queueing Theory (MAE634)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE634
Semester	6
Course Title	Queueing Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 course learning outcomes are: the study and development models that describe and analyse the behaviour and performance of queueing systems and their applications for optimal decision making. Upon successful completion of the course the student will be able to: recognize and implement M/M/1 queue model and its variants apply the Little's result recognize and implements M/G/1 queue model apply Markov processes to model queueing systems apply queueing models for decision making.

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. Birth death process. Transforms. Markovian Queueing Systems (Μ/Μ/1/∞, Μ/Μ/m/k, Μ/Μ/m/m, Μ/Μ/∞/∞). Queue with group arrival, Queue with group services, M/G/1/∞. Applications for optimal decision making.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	- Software for the calculation of queueing systems performance measures, Email, class web
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 78 Fieldwork (3-4 set of homework) 33 Course total 150
Student Performance Evaluation	LANGUAGE OF EVALUATION: Greek METHODS OF EVALUATION: Final exam (100%)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Δ. Φακίνος. Ουρές Αναμονής. 2η Έκδοση, Εκδόσεις Συμμετρία, 2008 (Κωδ. Εύδοξου: 45392).
• Α. Οικονόμου. Θεωρία Ουρών Αναμονής [Προπτυχιακό εγχειρίδιο]. Κάλλιπος, Ανοικτές Ακαδημαϊκές Εκδόσεις, 2023. https://dx.doi.org/10.57713/kallipos-182(Κωδ. Εύδοξου: 121051698).
• Α. Σταφυλοπάτης, Γ. Σιόλας. Ανάλυση Επίδοσης Υπολογιστικών Συστημάτων [Προπτυχιακό εγχειρίδιο]. Κάλλιπος, Ανοικτές Ακαδημαϊκές Εκδόσεις, 2015.https://dx.doi.org/10.57713/kallipos-453(Κωδ. Εύδοξου: 59303597).
• I. Adan, J. Resing. Queueing Theory. Eindhoven. Notes available online https://www.win.tue.nl/jadan/queueing.pdf , 2001.
• J. Medhi. Stochastic Models in Queueing Theory, Academic Press, New York, 2003.
• P. Phuoc Tran-Gia, T. Hosfeld. Performance Modeling and Analysis of Communication Networks, 2017.
• [Περιοδικό / Journal] Queuing Systems (Springer)
• [Περιοδικό / Journal] Stochastic Models (Taylor - Francis)
• [Περιοδικό / Journal] European Journal of Operational Research (Elsevier)

Numerical Analysis (MAE642)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ642
Semester	6
Course Title	Numerical Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 the basic theory of orthogonal polynomials, be aware and apply the taught methods of numerical integration be aware and apply the taught methods for numerical solution of equations and nonlinear systems, 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

Sets of Orthogonal Polynomials: Legendre, Chebyshev. Numerical Integration: Newton-Cotes, Chebyshev, Gauss-Legendre, Gauss-Chebyshev. Numerical Solution of Equations: Newton's Method, Secant Method, Aitken-Steffensen Methods. Numerical Solution of Nonlinear Systems: Newton's Method.

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 bibliografy 104 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Written examination

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• "Introduction to Numerical Analysis". Akrivis G.D., Dougalis B.A, Crete University Press, 4th Edition, 2010.

Introduction to Symbolic Mathematics (MAE644)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE644
Semester	6
Course Title	Introduction to Symbolic Mathematics
Independent Teaching Activities	Lectures and laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)
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 course is an introduction to symbolic mathematical computations (computer algebra) and programming using a language for processing symbolic mathematical expressions, such as Mathematica. The course examines basic concepts in symbolic algebraic computations and emphases is given on finding the solution of a problem in closed form (exact solution) as opposed to a numerical solution (approximate solution). Using a symbolic language the course examines tools / commands to solve problems from different areas of Mathematics (Calculus, Algebra, Geometry, Statistics, etc.) and how to graphically show the results of solving a problem. Also programming methods are examined which can be used for the solution of a problem in addition to using just ready commands. Much of the course is to present the possibilities and tools available in a programming language for symbolic processing of mathematical expressions. After completing the course the student: Has an understanding of the basic concepts of the symbolic processing of mathematical expressions. Can use software packages for symbolically processing mathematical expressions and design/implement procedures using these packages for solving a problem in a closed form. Can present and explain the solution to a problem using graphics.

General Competences

Working independently Teamwork Analysis of Problem Data Can use a computer algebra programming language to solve a problem and if possible to visualize data and solution. May solve problems in various disciplines with appropriate mathematical modeling.

Syllabus

1. Symbolic mathematical manipulation systems
2. Introduction to Mathematica
3. Representation of symbolic mathematical expressions
4. Numerical computations
5. Symbolic computations
6. Symbolic manipulation of mathematical expressions
7. Basic functions of Mathematica
8. Lists
9. Patterns and transformation rules
10. Input / Output and Files
11. Functions
12. Structures for program flow control (assignment, selection, loops, etc)
13. Programming with Mathematica
14. Graphics
15. Factorization
16. Solving equations and systems
17. Differentiation
18. Integration
19. Series
20. Linear algebra
21. Basic algorithms in symbolic mathematics

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, projects 33 Course total 150
Student Performance Evaluation	Written final exam (70%) comprising: questions about the processing of symbolic mathematical expressions using programming languages for this purpose Term project (teams) (30%) students in groups do a term project which basically consists of using Mathematica to work on a specific mathematical topic (presentation of concepts, problem solving, etc.)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• SCHAUM'S MATHEMATICA, EUGENE DON, 2006, Publicer KLEIDARITHMOS (translation)
• Mathematics and programming with Mathematica, Karampetakis Nikolaos, Stamatakis Stylianos, Psomopoulos Evangelos, 2004, Publicer Ziti Pelagia & Co.
• Wolfram, S., The Mathematica Book, 5 Edition, Wolfram Media.
• Abell, M., Braselton, J., Mathematica by Example, 2d Edition, Academic Press, 1997.
• Gaylord, R., Kamin, S., Wellin, P., An Introduction to Programming with Mathematica, 2d Edition, Telos Springer-Verlag, 1996.
• Gray, J., Mastering Mathematica - Programming Methods and Applications, 2d Edition, Academic Press, 1998.
• http://www.wolfram.com/
• http://library.wolfram.com/

Techniques of Mathematical Modelling (MAE646)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE646
Semester	6
Course Title	Techniques of Mathematical Modelling
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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, for example, the analysis of runtime algorithms. 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

Search for, analysis and synthesis of data and information, with the use of the necessary technology. Adapting to new situations. Decision-making.

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	Face to face
Use of Information and Communications Technology	Yes
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Exercises 33 Course total 150
Student Performance Evaluation	Weekly homework Final project Final exam

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, Springer, 1999.
• E. J. Hinch, Perturbation Methods, Cambridge University Press, 1991.
• A. H. Nayfeh, Perturbation Methods, Wiley-Interscience, 1973.

Object Oriented Programming (MAE647)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE647
Semester	6
Course Title	Object Oriented Programming
Independent Teaching Activities	Lectures, laboratory exercises, tutorials, quiz (Weekly Teaching Hours: 3, Credits: 6)
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 aims at introducing to students basic concepts and techniques related to object oriented programming. Introduction to object oriented programming, the notions of classes and objects in programming, Abstraction, Encapsulation, Modularity, Hierarchy. After successfully passing this course the students will be able to: Understand basic programming techniques Analyze complex programmes Develop software systems that are valuable, reliable, and flexible.

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

• Introduction to object oriented programming
• Classes and objects in programming
• Properties and methods
• Simple and multiple inheritance
• Abstraction
• Encapsulation
• Modularity
• Hierarchy and Composition

Teaching and Learning Methods - Evaluation

Delivery	Lectures
Use of Information and Communications Technology	Use of projector and interactive board during lectures. Course website maintenance. Announcements and posting of teaching material (lecture slides and notes, programs). Announcement of assessment marks via the ecourse platform by UOI.
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Team work 33 Course total 150
Student Performance Evaluation	Final written examination (70%) Exercises (30%)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Software Engineering - Theory & Practice, S. L. Pfleeger, ISBN 978-960-461-477-6
• Software Engineering, I. Sommerville, ISBN 978-960-461-220-8
• Βασικές Αρχές Γλωσσών Προγραμματισμού, Ellis Horowitz, Εκδόσεις Κλειδάριθμος

ICT in Education (MAE649)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE649
Semester	6
Course Title	ICT in education
Independent Teaching Activities	Lectures, laboratory exercises, tutorials, quiz (Weekly Teaching Hours: 3, Credits: 6)
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	-
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

ICT as a teaching and learning tool. Basic concepts and didactic tools of Informatics, Internet and educational applications (HTML, JavaScript), Learning Management Systems and tools (LMS, OBS studio-Twitch TV, Jitsi, Zoom), interactive educational technologies (MIT scratch), Multimedia applications programming for educational purposes (Adobe Flash), computational educational tools, educational tools for Mathematics (Geogebra, MathML, Maxima), mobile, IoT and werable educational technologies (BLE, Wi-Fi, Beacons, NFC, touchpad, Android studio, tinkercad, circuits simulator-3D printing), mathematical word processing tools (LateX), image and video processing tools (Gimp, Audacity, SynFig Studio, Blender, Tupitube), programming of mobile educational, tactile, remote surveillance and feedback applications using Blynk.

Teaching and Learning Methods - Evaluation

Delivery	Lectures
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Team work 33 Course total 150
Student Performance Evaluation	Final written examination (70%) Exercises (30%)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Data Structures (MAE681)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE681
Semester	6
Course Title	Data Structures
Independent Teaching Activities	Lectures and laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)
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 course is an introduction to basic data structures such as strings, arrays, lists, stacks, queues, trees, graphs. It studies properties and implementation issues as well as basic properties on the data structures and their complexity. It also examines basic applications of the above data structures. The main purpose is the design and use of appropriate data structures for storing and retrieving the data of a problem in order for a most efficient processing during the problem solving process. After completing the course the student: Has an understanding of basic data structures and the different ways they can be implemented using a programming language. Can choose appropriate data structures for efficiently storing the data of a problem and their use by an algorithms for solving the problem.

General Competences

Working independently Problem data analysis Can use data structures for solving problems in other scientific areas or in the workplace.

Syllabus

• Elements Of Analysis Of Algorithms
• Abstract Data Types
• Strings
• Arrays
• Algorithms for Searching, Sorting, Selection
• Lists (Single Linked Lists, Doubly Linked Lists, Circular Lists, Generalised Lists)
• Stacks
• Queues, DeQueues, Priority Queues
• Trees (General Trees, Binary Trees, Binary Search Trees, Threaded Trees)
• Heaps
• AVL-Trees, 2-3 Trees, 2-3-4 Trees, B Trees
• Directed Graphs, Undirected Graphs
• Set Manipulation
• Hashing
• Dynamic Memory Management

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 33 Course total 150
Student Performance Evaluation	Written final exam (70%) comprised of: questions about the theory of data structures Questions crisis in the form of exercises that require the use of data structures Exercises testing the understanding of the implementation issues and use of data structures Laboratory exercises / midterm (30%)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Data structures, algorithms and applications using c ++, Sahnii Sartaj, Publicer A. Tziola (Greek translation)
• Algorithms in C ++, parts 1-4: fundamental concepts, data structures, sorting, searching, Robert Sedgewick, Prentice Hall (Greek translation)
• Algorithms in C, parts 1-4: fundamental concepts, data structures, sorting, searching, Robert Sedgewick, Prentice Hall (Greek translation)
• Data Structures with C, Nicholas Misirlis (Greek)
• Data Structures, Bozanis Panagiotis, Publicer A. Tziola (Greek)
• Michael T. Goodrich, Roberto Tamassia, and David M. Mount, Data Structures and Algorithms in C ++, John Wiley & Sons
• Michael Goodrich, Roberto Tamassia, Data Structures and Algorithms in Java, Publicer DIAYLOS
• Cormen, Leiserson and Rivest, Introduction to Algorithms, MIT Press, 1990. (there is also a translation from the University of Crete)
• 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/
• http://opendatastructures.org/

Numerical Linear Algebra (MAE685)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ685
Semester	6
Course Title	Numerical Linear Algebra
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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

Upon successful completion of this course, students will be able to: describe and apply numerical methods from a variety of linear algebra problems. recognize the limitations of finite precision arithmetic in calculations and explain the importance of the stability of numerical algorithms. evaluate numerical methods for their accuracy, efficiency, and applicability. implement in Octave or Python numerical algorithms and apply appropriate criteria to terminate an iterative algorithm.

General Competences

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

Syllabus

• Introduction to matrix theory. Singular Value Decomposition (SVD). Matrix condition number and conditioning of linear systems.
• The linear least squares problem, QR method, Householder transformations.
• Direct methods (LU Factorization, Cholesky Factorization).
• Iterative methods: Jacobi, Gauss-Seidel, SOR method, steepest descent method, conjugate gradient method.
• Computation of eigenvalues ​​and eigenvectors.
• Applications (PageRank Google search algorithm, image processing, etc.)

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 e-learning platform. Provision of study materials in Moodle e-learning platform. Provision of model solutions for some exercises in podcast format. Communication with students through e-mails, Moodle platform and Microsoft Teams. IT sessions (Python or Octave) for the implementation of the numerical algorithms.
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 76 Directed study of exercises 5 Exercises-Homeworks 30 Course total 150
Student Performance Evaluation	Computer-based exercises with oral examination (Weighting 30%, addressing learning outcomes 2-4) Written examination (Weighting 100%, addressing learning outcomes 1-3)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• “Αριθμητική Γραμμική Άλγεβρα”, Β. Δουγαλής, Δ. Νούτσος, & Α. Χατζηδήμος, Τυπογραφείο Πανεπιστημίου Ιωαννίνων.
• “Numerical Linear Algebra”, L. Trefethen, & D. Bau, SIAM, 1997.
• “Matrix Computations”, G. Golub, C. Van Loan, 3rd edition, Johns Hopkins Univ. Press 1996.
• “Iterative Methods for Sparse Linear Systems”, Y. Saad, PWS Publishing, 1996.
• “Linear Algebra and Learning from Data”, G. Strang, Wellesley-Cambridge Press, 2019.
• “Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control”, S. Brunton, & J. Kutz, Cambridge: Cambridge University Press, 2019. doi:10.1017/9781108380690.

Complex Functions II (MAE712)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ754
Semester	7
Course Title	Seminar in Analysis II
Independent Teaching Activities	Lectures (Weekly Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	None. However it is desirable to have a strong knowledge of basic notions of differential equations.
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	No
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The ability to write a complete report on a scientific subject. The ability to present this report. Working in teams. The report can be, but not required to be, original. Further details can be determined by the teaching professor.
General Competences	Αnalyse and combine data and information using various technologies. Working independently and in groups. Free, creative, analytic, and conclusive thinking. Decision making. Further details can be determined by the teaching professor.

Syllabus

In depth study in a scientific subject related to mathematical analysis. Further details can be determined by the teaching professor.

Teaching and Learning Methods - Evaluation

Delivery	Details will be determined by the teaching professor. Methods include presentations contacted by the students.
Use of Information and Communications Technology	Details will be determined by the teaching professor.
Teaching Methods	Activity Semester Workload Study in class 39 Other activities determined by the teaching professor 111 Course total 150
Student Performance Evaluation	There is no final exam. Each student must write a report on a specific subject. Each student must present the report publically. Students may miss up to 3 lectures. Other means of evaluation can be determined by the teaching professor.

Attached Bibliography

Bibliography is suggested by the teaching professor, depending on the subject under study.

Partial Differential Equations (MAE713)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE713
Semester	7
Course Title	Partial Differential Equations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, English
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 an introduction to Partial Differential Equations. By this course the students become familiar with a broad area of Analysis that has many applications to other sciences. The course highlights the wealth of problems that arise and proposes methods to overcome them. These are presented exemplarily and aim to teach ways of transcending and generalizing known methods and solutions. The students learn to analyze methodically externally given problems, taking into account relevant informations and aims, and to try to apply knowledge from other areas of Pure Mathematics in order to solve these problems. Moreover, the students learn to interpret the obtained mathematical results. On the level of content, the students learn about, mainly linear, Partial Differential Equations of first and second order for functions of two variables with respect to both, their explicit solution and their qualitative behavior, and obtain an elementary overview of further problems.

General Competences

Search for, analysis and synthesis of data and information, with the use of the necessary technology Working independently Working in an interdisciplinary environment Production of free, creative and inductive thinking

Syllabus

• Overview of Partial Differential Equations (PDE) and Systems: classification with respect to their (non-)linearity, description of the arising problems and of the various kinds of solutions (classical and weak, general and with boundary values).

(In the following the focus is given on two independent variables.)

• First order PDE (linear, semi-linear, quasi-linear): geometric and algebraic observations concerning their qualitative behavior, initial value problems and method of characteristics, discussion of the Burgers equation, shock waves and weak solutions, Rankine-Hugoniot condition.
• Second order PDE: classification, characteristic directions and characteristic curves, wave equation on the line (homogeneous and inhomogeneous), separation of variables for the Laplace and heat equations, Poisson formula.

(Alternatively: instead of the discussion of the (non-linear) Burgers equation and of weak solutions an introduction to the Fourier transform may be given and the heat equation on the line may be discussed.)

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	The students may contact the lecturer by e-mail
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Written exam (mandatory) Homework (optional)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Δάσιος, Γ., Κυριάκη, Κ., & Βαφέας, Π. (2023). Μερικές Διαφορικές Εξισώσεις [Προπτυχιακό εγχειρίδιο]. Κάλλιπος, Ανοικτές Ακαδημαϊκές Εκδόσεις. http://dx.doi.org/10.57713/kallipos-317
• L. C. Evans. Partial Differential Equations. Second edition. AMS, 2010.
• G. B. Folland. Introduction to Partial Differential Equations. Princeton University Press, 1995.

Set Theory (MAE714)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE714
Semester	7
Course Title	Set Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 plan of the course is an introduction to Axiomatic Set Theory.
General Competences	Working independently Team work Production of free, creative and inductive thinking

Syllabus

The construction of the sets of numbers (Natural, Rational and Real numbers), Axioms for the Zermelo-Fraenkel theory, the Axiom of Choice, Zorn's Lemma, Well ordered sets, Ordinal and Cardinal Numbers and arithmetic of them.

Teaching and Learning Methods - Evaluation

Delivery	Lectures\ Presentations in class
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Assignments/Essays 33 Individual study 78 Course total 150
Student Performance Evaluation	Written examination at the end of the semester.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Derek Goldrei, Classical Set Theory
• Γ. Μοσχοβάκη, Θεωρία Συνόλων
• R. Vaught, Set Theory, An Introduction
• Paul Halmos, Naïve Set Theory

Ordinary Differential Equations II (MAE716)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE716
Semester	7
Course Title	Differential Equations I
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special 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

Bloom's Taxonomy. (1) Remembering: The notion of functional differential equation, of integral equation, of integral-differential equation and of difference equation. The notion of solutions of such equations, of uniqueness of such solutions and of stability of such solutions. The notion of solutions of systems of difference equations. (2) Comprehension: Study of solutions of functional ODE's, of integral equations and of difference equations. Methods of finding such solutions and of studying their stability. Study of systems of such equations. (3) Applying: Study related real world problems. (4) Evaluating: Teaching secondary school courses.

General Competences

Working independently and in groups. Production of free, creative and inductive thinking. Creative, analytic and synthetic thinking.

Syllabus

Section 1. Functional differential equations: Reasons of existence of such equations, Existence and uniqueness of their solutions, Finding solutions, Stability, Linear and non-linear systems. Section 2. Integral equations: Reasons of existence of such equations, Fredholm equations, Volterra equations, Integral-difference equations, Abel problem, Non-linear integral equations. Section 3. Difference equations: Reasons of existence of such equations, Finding the formula of solutions for linear difference equations, Linearization, Systems of difference equations, Stability using the Lyapunov method.

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, 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 (7x3) 21 Seminars (6x3) 18 Individual study 78 Exrecises/projects 33 Course total 150
Student Performance Evaluation	Language of evaluation: Greek and English. Methods of evaluation: Weekly written assignments. Few number of tests during the semester. Based on their grades in the aforementioned weekly assignments and tests, limited number of students can participate in exams towards the end of the semester, before the beginning of the exams period. In any case, all students can participate in written exams at the end of the semester, during the exams period. 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

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Harmonic Analysis (MAE718)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE718
Semester	7
Course Title	Harmonic Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 achievement by the undergraduate student of the theoretical background in the theory of Fourier series
General Competences	The objective of the course is the undergraduate student's ability achievement in analysis and synthesis of the basic background in Harmonic Analysis.

Syllabus

Trigonometric polynomials, partial sums of the Fourier series of a function, Bessel's inequality, Lemma Riemann-Lebesgue, Parseval's identity for Riemann integrable functions, complex Riemann integrable functions defined on an interval, Fourier coefficients and Fourier series, the Dirichlet kernel, criteria for uniform convergence of the Fourier series, convolution of functions and approximations to the identity, Fejer kernel, theorem of Fejer, Poisson kernel, Abel summability of the Fourier series, applications.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 78 Exercises solutions 33 Course total 150
Student Performance Evaluation	Written examination at the end of the semester.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Yitzhak Katznelson, An Introduction to Harmonic Analysis, Dover Edition.
• Elias M. Stein, Rami Shakarchi, Fourier Analysis, An Introduction, Princeton University Press.

Functional Analysis (MAE719)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE719
Semester	7
Course Title	Functional Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 goal of this course is: To familiarize the student with the notions, the basic theorems and the techniques concerning normed vector spaces, Banach spaces, Hilbert spaces, bounded linear operators between them and the dual spaces. After completing this course the student will be able to recognize if a given normed linear space is a Banach space, to compute the norm of a bounded linear operator, to use the basic theorems of Functional analysis (Hahn-Banach theorem and its consequences, Open mapping theorem, Closed graph theorem, Banach-Steinhaus theorem, Uniform Boundedness Principle).

General Competences

Αnalyse and combine data and information using various technologies. Working independently and in groups. Free, creative, analytic, and conclusive thinking. Decision making.

Syllabus

Linear spaces and algebraic bases (Hamel bases). Linear operators. Normed linear spaces. Banach spaces and classical examples. Finite dimensional spaces. Bounded linear operators, bounded linear functionals and computation of their norm. Dual space. Conjugate operators. Hahn Banach theorem and its consequences. The second dual space. Reflexive spaces. Baire's category theorem and some of its consequences in Functional Analysis (Open Mapping Theorem, Closed graph Theorem, Uniform Boundedness Principle, Banach-Steinhaus Theorem). Elements from Hilbert spaces.

Teaching and Learning Methods - Evaluation

Delivery	Teaching on the blackboard from the teacher
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Individual study 78 Solving exercises-homework 33 Course total 150
Student Performance Evaluation	Exams in the end of the semester (mandatory), intermediate exams (optional), assignments of exercises during the semester (optional).

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Special Topics in Algebra (MAE723)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE723
Semester	7
Course Title	Special Topics in Algebra
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 principal aim of the course is to introduce the students to the main ideas and methods of Commutative Algebra.
General Competences	The course promotes inductive and creative thinking and aims to provide the student with the theoretical background and skills of commutative rings.

Syllabus

• Polynomial Rings
• Hilbert's Basis Theorem
• Localization
• Integral dependence
• Hilbert Series
• Dimension
• Groebner Bases
• Hilbert's Nullstellensatz Theorem

Teaching and Learning Methods - Evaluation

Delivery	Teaching on the blackboard by the teacher.
Use of Information and Communications Technology	Communication with the teacher by electronic means (i.e. e-mail).
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Personal study 78 Solving exercises 33 Course total 150
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which includes resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• J.Beachy, Introductory Lectures on Rings and Modules, LMS, Cambridge University Press, (1999).
• D.Dummit, R.M.Foote, Abstract Algebra, 3 edition, Prentice Hall, (2003).
• N.Jacobson, Basic Algebra I & II, W. H. Freeman and Company, (1985 & 1989).
• S.Lang, Algebra, Graduate Texts in Mathematics, Springer (2002).
• L.Rowen, Ring Theory, Academic Press, 2 edition (1991).
• Μαλιάκας. Ταλέλλη, Πρότυπα πάνω από Περιοχές Κυρίων Ιδεωδών και Εφαρμογές, Εκδ. Σοφία (2009).
• Α. Μπεληγιάννης, Μια Εισαγωγή στη Βασική Άλγεβρα, Εκδ. Κάλλιπος (2015).

Algebraic Structures II (MAE724)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE724
Semester	7
Course Title	Algebraic Structures II
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 students will acquire with the successful completion of the course the skills to solve equations of small degree, the skills to find splitting fields and compute Galois groups, understand the problem of solving polynomial equations by radicals, understand the impossibility or not of certain constructions by ruler and compass.
General Competences	The course aim is for the student to acquire the ability in analysis and synthesis of knowledge in Field Theory and produces free, creative and inductive thinking.

Syllabus

• Rings
• Integral Domains, Fields, Homomorphisms and Ideals
• Quotient Rings
• Polynomial Rings over fields
• Prime and Maximal Ideals
• Irreducible Polynomials
• The classical methods of solving polynomial equations
• Splitting fields
• The Galois Group
• Roots of unity
• Solvability by Radicals
• Independence of characters
• Galois extensions
• The Fundamental Theorem of Galois Theory
• Discriminants
• Polynomials of degree less than 4 and Galois Groups
• Ruler and Compass constructions

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which includes resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• M. Holz: "Repetition in Algebra", Greek Edition, Symmetria Publishing Company, (2015).
• Th. Theochari-Apostolidou and C. M. A. Charalambous:  "Galois Theory", (Greek), Kallipos Publishing (2015).

Ring Theory (MAE725)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE725
Semester	7
Course Title	Ring Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, 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

The principal aim of the course is to introduce the students to the main tools and methods of the theory of non-commutative rings, where by non-commutative ring is meant an associative ring with unit, which is not necessarily commutative. The main objective of the course is to present the basic theory of rings and the ideas which lead to the proof of: (a) the fundamental theorem of Wedderburn-Artin concerning the structure of semisimple rings and, (b) the fundamental density theorem of Jacobson concerning the structure of primitive rings. A key element in the study of a ring is the interaction and interplay between ring-theoretical properties of the ring and the structure of its (left or right) ideals or modules. In the course a variety of examples and constructions will be analyzed and various applications of ring theory to other areas of mathematics (in particular of algebra) will be explored. At the end of the course we expect the student to have understood the definitions and basic theorems which are discussed in the course, to have understood how they are applied in discrete examples, to be able to apply the material in order to extract new elementary conclusions, and finally to perform some (no so obvious) calculations.

General Competences

The course aims to enable the undergraduate student to acquire the ability to analyse and synthesize basic knowledge of the Theory of Rings, which is an important part of modern algebra. The contact of the undergraduate student with the ideas and concepts of the theory of rings, (a) promotes the creative, analytical and deductive thinking and the ability to work independently, (b) improves his critical thinking and his ability to apply abstract knowledge in various field.

Syllabus

Rings - Homomorphisms - Ideals - Quotient Rings - Modules - Rings arising from various constructions - Algebras - Group algebras - Modules over group algebras - Module homomorphisms - The bicommutator - Simple faithful modules and primitive rings - Artin rings - Simple finite dimensional algebras over algebraically closed fields - Artinian modules - Noetherian rings and modules - Jacobson radical.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face to face)
Use of Information and Communications Technology	Teaching Material: Teaching material in electronic form available at the home page of the course. Communication with the students: Office hours for the students (questions and problem solving). Email correspondence Weekly updates of the homepage of the course.
Teaching Methods	Activity Semester Workload Lectures (13x3) 39 Working independently 78 Exercises - Homeworks 33 Course total 150
Student Performance Evaluation	Combination of: Weekly homework, presentations in the class by the students, written work, and, at the end of the semester, written final exams in Greek (in case of Erasmus students, in English) which includes analysis of theoretical topics and resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Nathan Jacobson: "Basic Algebra I & II", W. H. Freeman and Company,  (1985 & 1989). 
• I.N. Herstein: "Non-commutative Rings", AMS, Carus Mathematical Monographs 85, (1971).
• Luis Rowen: "Ring Theory (student edition)", Academic Press, Second Edition, (1991).
• T.Y. Lam: "A First Course in Noncommutative Rings", GTM 131, Springer, (2001).
• P. M. Cohn: "Introduction to Ring Theory", Springer (2000).
• Y. Drozd and V. Kirichenko: "Finite Dimensional Algebras", Springer (1994).

Euclidean and Non Euclidean Geometries (MAE727)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE727
Semester	7
Course Title	Euclidean and Non Euclidean Geometries
Independent Teaching Activities	Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, 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	This is an introductory course on non Euclidean geometries. The aim is to study how the attempt to prove Euclid's fifth postulate led the way to non Euclidean geometries. On completion of the course the student should be familiar with the foundations of Euclidean and non Euclidean geometries.
General Competences	Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism

Syllabus

Euclid's geometry, Hilbert's system of axioms, the fifth postulate, compatibility of axioms, neutral geometry, independence of the fifth postulate, hyperbolic geometry, Poincarẻ model, spherical geometry, Platonic solids.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which includes resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Π. Πάμφιλου, Γεωμετρία, Εκδόσεις Τροχαλία, 1989.
• M.J. Greenberg, Euclidean and non-Euclidean Geometry-Development and History, W.H. Freedmann and Company, 1973.
• R. Hartshorne, Geometry: Euclid and beyond, Springer, 2000.
• H. Meschkowski, Noneuclidean Geometry, Academic Press, 1964.

Differentiable Manifolds (MAE728)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE728
Semester	7
Course Title	Differentiable Manifolds
Independent Teaching Activities	Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, 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	In this lecture we introduce basic notions of modern Differential Geometry. More precisely, we introduce among others the notions of manifold, tangent bundle, connection, parallel transport and Riemannian metric.
General Competences	work autonomously work in teams develop critical thinking skills.

Syllabus

• Smooth manifolds.
• Smooth maps.
• Tangent vectors.
• Vector fields.
• Regular values and Sard's Theorem.
• Homotopy and Isotopy.
• Lie bracket.
• Frobenius' Theorem.
• Connections and parallel transport.
• Riemannian metrics.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Autonomous Study 111 Course total 150
Student Performance Evaluation	Weakly homeworks and written final examination.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• M. do Carmo, Riemannian Geometry, Birkhaüser Boston, Inc., Boston, MA, 1992.
• V. Guillemin & A. Pollack, Differentiable Topology, Prentice-Hall, Inc, Englewood Cliffs, 1974.
• J. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218, 2013.
• J. Milnor, Topology From the Differentiable Viewpoint, Princeton University Press, NJ, 1997.
• L. Tu, An Introduction to Manifolds, Universitext. Springer, New York, 2011.
• Δ. Κουτρουφιώτης, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων, 1994.

Decision Theory - Bayesian Theory (MAE731A)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ731A
Semester	7
Course Title	Decision Theory - Bayesian Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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

This course consists of two modules: the Decision Theory and Bayes Theory. The Decision Theory deals with problems of decision-making. Object of Statistical Decision Theory is decisions about unknown numerical quantities (parameters) by utilizing the presence of statistical knowledge. The aim of the course is the evaluation of the performance of the estimators subject to properties such as the unbiasedness, sufficiency, consistency etc. The second part of the course gives an introduction to Bayesian statistical approach. At the end of the course the student should be able to compare Bayes and classical approaches and evaluate the "performance" of different estimators by using various criteria.

General Competences

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

Syllabus

Decision Theory: decision function, loss function, risk function, admissible and minimax estimators; Bayesian inference: Bayes estimators, Bayes confidence intervals, minimax and Bayes tests.

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-Homeworks 33 Course total 150
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which concentrates on the solution of problems which are motivated by the main themes of the course.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Berger, J.O. (1985) Statistical decision theory and Bayesian analysis. Springer.
• Bernardo J. M. & Smith A. F. M., (1994). Bayesian Theory, Wiley, London.
• Congdon, P. (2007), Bayesian Statistical Modelling, Willey.
• Κ. Φερεντίνος (2005). Εκθετική οικογένεια κατανομών Θεωρία Bayes, Πανεπιστημιακές Παραδόσεις.

Topics in Operations Research (MAE732A)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE732A
Semester	7
Course Title	Topics in Operations Research
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 course learning outcomes are: the introduction of the students to integer programming formulations, the introduction of the students to the dynamic programming methodology, the introduction of the students to techniques and tools for decision-making under uncertainty. Upon successful completion of the course the student will be able to: model and solve integer programming problems and understand their differences with the linear programming problems. understand the basic principles of dynamic programming construct simple recursive dynamic programming equations solve known optimization problems using dynamic programming describe and handle decision making problems under uncertainty.

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

Integer linear programming (integer and mixed integer problems formulation, integer programming algorithms). Dynamic programming (Bellman principle of optimality, finite and infinite horizon problems, Applications on: Routing problems, Equipment-Replacement Problem, inventory problems, etc). Decision analysis (General characteristics of decision problems, decisions under uncertainty, decision trees, risk analysis).

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	Lindo/Lingo Software, Email, class web
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 78 Fieldwork (3-4 set of homework) 33 Course total 150
Student Performance Evaluation	LANGUAGE OF EVALUATION: Greek METHODS OF EVALUATION: Final exam (100%)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Bellman, R.E.. Dynamic Programming, Princeton University Press, 1957, Princeton, NJ. Republished 2003
• Bertsekas D. P. Dynamic Programming and Optimal Control, Vols. I and II, Athena Scientific, 1995, (3 Edition Vol. I, 2005, 4th Edition Vol. II, 2012),
• BERTSIMAS D. and J. N. TSITSIKLIS Introduction to Linear Optimization, Athena Scientific 1997
• HADLEY G. Linear Programming, Addison-Wesley Publishing Company, INC, 1965
• HILLIER F. S. and G. J. Lieberman. Introduction Operations research. The McGraw-Hill Companies, 2001
• WINSTON W. L., Operations research (Applications and algorithms). Duxbury Press (International Thomson Publishing) 1994.
• [Περιοδικό / Journal] Mathematical Programming Journal, Series A and Series B
• [Περιοδικό / Journal] INFORMS Transactions on Education (ITE) 

Regression and Analysis of Variance (MAE733)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ733
Semester	7
Course Title	Regression and Analysis of Variance
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 aim of the course is the presentation, study and application of linear models and more precisely the simple and multiple linear regression models and analysis of variance of one or more factors, as well. The general linear model is presented to unify the above mentioned regression and analysis of variance models. This course is focused on the theory of linear models and their applications in modelling statistical data. At the end of the course, students understand the aforementioned issues of the theory of linear models and it is, moreover, expected that they will be able to apply the theory of linear models for the analysis of real statistical data.

General Competences

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

Syllabus

Theory of linear models. Simple linear regression. Multiple linear regression. One-and multi-way analysis of variance. Multiple comparisons. 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-Homeworks 33 Course total 150
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which concentrates on the solution of problems which are motivated by the main themes of the course.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Kutner, M. H., Nachtsheim, Ch., Neter, J. and Li. W. (2004). Applied Linear Statistical Models. 5 Edition, McGraw-Hill.
• Montgomery, D. C., Peck, E. A. και Vining, G. G. (2006). Introduction to linear regression analysis. 4th Edition, Wiley.
• Rencher, A. C. (2000). Linear models in statistics. Wiley.
• Sahai, H. and Ageel, M. (2000). The Analysis of Variance. Birkhauser.
• Καρακώστας, Κ. (2002). Γραµµικά Μοντέλα: Παλινδρόµηση και Ανάλυση ∆ιακύµανσης. Πανεπιστήµιο Ιωαννίνων.

Database Systems and Web Applications Development (MAE741)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE741
Semester	7
Course Title	Database Systems and Web applications development
Independent Teaching Activities	Lectures-Laboratory (Weekly Teaching Hours: 3, Credits: 6)
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

Students knowledge acquisition of design, implementation procedures and methodologies using Relational DataBase Management Systems (RDBMS), as well as familiarity with the development of Internet programming applications using PHP, JavaScript, jQuery and AngularJS. Basic Internet programming concepts HTML, CSS, Database relationships, tables and structure. Concepts and architecture of Database Systems, Relational model, Internet programming languages ​​and Tier System architecture. Data Modelling using the relational SQL Database Language MariaDB, SQL Queries, Normalization, Normal forms, Non relational Databases - MongoDB. Databases on the Internet using programming languages, PHP programming language, using PHP for mathematical problems, using MySQLi Api, Bootstrap, Introduction to JavaScript, AJAX, JSON and jQuery.

General Competences

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

Syllabus

1. Data models with emphasis on relational model. Introduction to relational algebra and relational calculus. Conceptual Models: Entity-Associations Model. Theory of dependencies. Form normalization (1NF, 2NF, 3NF, BCNF). Database design. Introduction to Database Management Systems.
2. SQL language with practical application using MariaDB. Create tables, modify fields, add records to a table, Database tables management.
3. Create basic SQL queries in MariaDB tables.
4. SQL joins, SQL table associations-relations, foreign keys, stored procedures, triggers.
5. Introduction to the web and its capabilities. Web page development. Basic HTML content formatting commands, Add images, create tables, lists and frames, HTML layers, divs HTML 5 additional commands.
6. HTML and content formatting using Cascading Style Sheets (CSS). Advanced ways of responsive formatting using the Bootstrap library.
7. Introduction to JavaScript, ways to import JavaScript into HTML, JavaScript DOM, functions and classes.
8. Introduction to PHP, basic language capabilities, input output, data types, conditions, repetitive loops.
9. Create forms in HTML and retrieve form information using PHP and JavaScript (AJAX), using GET, POST methods.
10. Use of PHP and MySQL, presentation of PHP input functions and retrieval of information from DB tables. (mysqli-PDO api). Creating dynamic web pages.
11. Mathematical extensions of PHP, PHP and data processing from DB to solve linear equation problems, presentation of the PHP-LAPACK class.
12. Mathematical extensions of PHP, PHP and statistical data processing from DB, presentation of PHP statistical functions.
13. Asynchronous communication with DB, PHP and AJAX, using the jQuery library and JSON configuration. Presentation and use of AngularJS and NodeJS frameworks.

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 33 Course total 150
Student Performance Evaluation	Using new ICT and metrics of the asynchronous e-learning platform (5%) Examination of laboratory exercises (10%) Semester work and written examination

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• PHP 6 AND MYSQL 5 FOR DYNAMIC WEB SITES, 5 Edition, LARRY ULLMAN, ISBN-13: 978-0134301846, 2018.
• JAVASCRIPT & JQUERY interactive front-end web development, Jon Duckett, ISBN-13: 978-1118531648, 2017.

Introduction to Computational Mathematics (MAE742A)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE742A
Semester	7
Course Title	Introduction to Computational Mathematics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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

Science is based on two major pillars, both theoretical and experimental. However, over the last few decades scientific computing has emerged and recognized as the third pillar of science. Now, in most scientific disciplines, theoretical and experimental studies are linked to computer analysis. In order for the graduate student to be able to stand with claims in the modern scientific and work environment, knowledge in computational techniques is considered a necessary qualification. The course aims to introduce the student into the field of computational mathematics, emphasizing the implementation of numerical methods using computers. The student will be able to familiarize himself with Matlab and Python programming languages, the most widespread for performing scientific calculations. Working autonomously and in groups, the student will be required to implement computational methods related to the fields of numerical analysis and numerical linear algebra. Specifically, the objectives of this laboratory course are: Familiarity with Matlab and Python programming languages to implement numerical methods and graphical design of the numerical solutions Implementation of polynomial interpolation and function approximation Apply numerical integration Solving linear and nonlinear equations Solving systems of linear equations Study of direct and iterative methods.

General Competences

The course aims to enable the student to: Search, analyze and synthesize data and information, using the available technologies Work autonomously Work in a team Promote free, creative and inductive thinking

Syllabus

• Vector and matrix definition and calculations
• Basic commands and functions
• Graphic representation of the numerical results
• Polynomial interpolation: Lagrange Method, Newton's Method
• Numerical integration: Simple and generalized types of numerical integration, rectangular rule, trapezoid rule, Simpson rule, Gauss integration
• Numerical solution of non-linear equations: iterative methods, bisection method, fixed point method, Newton's method
• Numerical solution of linear systems - Direct methods: Gauss elimination, LU decomposition.

Teaching and Learning Methods - Evaluation

Delivery	In the laboratory
Use of Information and Communications Technology	Use of scientific computing software packages
Teaching Methods	Activity Semester Workload Lectures 39 Study of bibliography 39 Laboratory exercises 39 Home exercises (project) 33 Course total 150
Student Performance Evaluation	Weekly assignments Final project Written examination at the end of the semester

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Introduction to Numerical Analysis, G.D. Akrivis, V.A. Dougalis, 2010 (in Greek).
• Numerical Linear Algebra, V. Dougalis, D. Noutsos, A. (in Greek).
• A Primer on Scientific Programming with Python, H. P. Langtangen, Springer-Verlag Berlin Heidelberg, 5 Edition, 2016.
• Programming for Computations- MATLAB/Octave, S. Linge, H. P. Langtangen, Springer International Publishing, 2016 (in Greek).

Introduction to Mathematical Physics (MAE743)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ743
Semester	7
Course Title	Introduction to Mathematical Physics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 analytic and numerical methods of Mathematical Physics. The objectives of the course are: Development of the theoretical background in matters relating to mathematical physics. Ability of the student to apply the basic concepts of mathematical physics. Upon completion of this course the student will be able to solve with analytical and approximate mathematical methods simple problems of mathematical physics and deepen further understanding of such methods.

General Competences

The course aims to enable the undergraduate students to develop basic knowledge of Mathematical Physics and in general of Applied Mathematics. The student will be able to cope with problems of Applied Mathematics giving the opportunity to work in an international multidisciplinary environment.

Syllabus

Short introduction of linear vector spaces, Vector spaces of infinite dimensions, The Sturm-Liouville problem, Orthogonal polynomials and special functions, Multi-dimensional problems, Operator Theory, Applications in modern Physics.

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 Study of theor 78 Home exercises 33 Course total 150
Student Performance Evaluation	Weekly assignments Final project Written examination at the end of the semester

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Numerical Solution of Ordinary Differential Equations (MAE744)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE744
Semester	7
Course Title	Numerical Solution of Ordinary Differential Equations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 the course, students will be able to: describe the basic characteristics of single-step and multi-step methods and recognize their differences. apply a variety of techniques for the construction of single-step and multi-step methods. apply numerical analysis techniques to show consistency, stability, and convergence of numerical methods. be aware of the optimal order of accuracy of key numerical methods as well as the limitations that may be required in the discretization parameters to ensure stability. write code (in Python or Octave) for the implementation of explicit and implicit numerical methods and calculate their experimental order of convergence. write code in Python or Octave for the numerical approximation of the solution of ODEs models that describe a variety of real-world situations.

General Competences

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

Syllabus

1. Initial Value Problems
2. Explicit Euler and Implicit Euler.
3. Consistency, stability, and convergence of Runge-Kutta methods.
4. Consistency, stability, and convergence of multistep methods.
5. Applications to ODEs systems arising from Physics and Biology.

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 e-learning platform. Provision of study materials in Moodle e-learning platform. 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) for the implementation of the numerical algorithms.
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 75 Directed study of Computer-based exercises 6 Non-directed study of Computer-based exercises 30 Course total 150
Student Performance Evaluation	Computer-based exercises (organised in teams of 2) with oral examination (Weighting 30%, addressing learning outcomes 4-6) Written examination (Weighting 70%, addressing learning outcomes 1-4)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• “Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations”, E. Hairer, & C. Lubich, Springer, 2010.
• “Numerical Methods for Ordinary Differential Equations: Initial Value Problems”, D.F. Griffiths, & D. J. Higham, Springer, 2010.

Theory of Computation (MAE745)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE745
Semester	7
Course Title	Theory of Computation
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 goal of this course is the deeper understanding of Automata Theory and Languages. During the course a detailed examination of the following topics are done: Introductory concepts of Automata , Computability and Complexity as well as basic definitions, basic theorems and inductive proofs Finite State Machines and Languages, Finite Automata (Deterministic FA, Nondeterministic FA, FA with Epsilon-Transitions) and their applications, Regular Expressions and Languages, derivation trees. Removing Nondeterminism . Equivalence NFA and NFA with ε-moves. Minimization of DFA, Pumping Lemma FA and Grammars. Grammars of Chomsky Hierarchy. Regular Sets (RS). Properties of Regular Languages. RS and FA. Finding a correspondence Regular Expression of a FA. Abilities and disabilities of FA. Context-Free Grammars and Languages, Pushdown Automata (Deterministic PDA, Acceptance by Final State, Acceptance by Empty Stack) , Properties of Context-Free Languages. Correspondence PDA and Context-Free Languages. Introduction of Turing Machines. Standard TM, useful techniques for TM constructions. Modification of TM. TM as procedure. Unsolvability. The Church-Turing Thesis. The Universal TM. The Halting Problem for TM. Computational Complexity. NP-complete problems. Upon completion of the course, the students will be able to: understand theoretical documentation of mathematical problems solve exercises track further applications which are related to Finite Automata, Pushdown Automata, and Turing Machines as well as to Unsolvability , to Computational Complexity and to NP-complete problems.

General Competences

Handle new problems Decision making Implementation- Consolidation

Syllabus

• Introduction and related concepts.
• Finite automata and regular expressions, regular languages, closure properties, pumping lemma, algorithms.
• Determinism, non-determinism.
• Pushdown automata and context-free grammars, context-free languages, closure properties, pumping lemma, algorithms.
• Chomsky normal form.
• Turing machines, equivalence of different models.
• Recursive and recursively enumerable languages.
• Church-Turing thesis.
• Undecidability, the halting problem, Post’s correspondence problem.
• Classes P and NP.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	Projector and interactive board during lectures. Course website maintenance. Announcements and posting of teaching material (lecture slides and notes, programs). Assessment marks via the ecourse platform.
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Exercises 33 Course total 150
Student Performance Evaluation	Final test

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• [11776]: Στοιχεία θεωρίας υπολογισμού, Lewis Harry R.,Παπαδημητρίου Χρίστος Χ.
• [257]: ΕΙΣΑΓΩΓΗ ΣΤΗ ΘΕΩΡΙΑ ΥΠΟΛΟΓΙΣΜΟΥ, SIPSER MICHAEL.

Graph Theory (MAE746)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE746
Semester	7
Course Title	Graph Theory
Independent Teaching Activities	Lectures, laboratory exercises, tutorials, quiz (Weekly Teaching Hours: 3, Credits: 6)
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

Introduction to fundamental concepts of graph theory and understanding of algorithmic techniques of graph problems. Basic definitions and concepts, Connectivity and Biconnectivity, Trees, Spanning Trees and Rooted trees, Eulerian and Hamiltonian graphs, Otpimization problems on graphs, Planar graphs, Graphs, connectivity, spanning trees, Eulerian & Hamiltonian graphs, Graph coloring, Clique, Independent set, Vertex cover, Planar graphs.

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

• Introduction to basic graph concepts
• Connectivity and biconnectivity
• Trees
• Eulerian & Hamiltonian graphs
• Graph optimization problems
• Planar graphs

Teaching and Learning Methods - Evaluation

Delivery	Lectures
Use of Information and Communications Technology	Use of projector and interactive board during lectures. Course website maintenance. Announcements and posting of teaching material (lecture slides and notes, programs). Announcement of assessment marks via the ecourse platform by UOI.
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Team work 33 Course total 150
Student Performance Evaluation	Final written examination (70%) Exercises (30%)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Γ. Μανωλόπουλος, Μαθήματα Θεωρίας Γράφων . Κωδικός Βιβλίου στον Εύδοξο: 3472
• Σημειώσεις στη Θεωρία Γραφημάτων, Χάρης Παπαδόπουλος, Πανεπιστήμιο Ιωαννίνων, 2012.
• Θεωρία γραφημάτων με παραδείγματα κ ασκήσεις, Κωδικός Βιβλίου στον Εύδοξο: 31528, Συγγραφείς: ΠΑΠΑΙΩΑΝΝΟΥ ΑΛΕΞΑΝΔΡΟΣ, Διαθέτης (Εκδότης): ΑΡΗΣ ΣΥΜΕΩΝ.

Linear and Nonlinear Waves (MAE747)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE747
Semester	7
Course Title	Linear and Nonlinear Waves
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 study of nonlinear systems has quietly and steadily revolutionized the realm of science over recent years. It is known that for nonlinear systems new structures emerge that have their features and peculiar ways of interacting. Examples of such structures abound in nature and include, amongst others: vortices (like tornadoes), solitons (bits of information used in optical fiber communications, water waves, tsunamis, etc), and chemical reactions. This course is intended as an introduction to the theory and of Nonlinear Waves and their applications. By the end of the course students will be able to: highlight the major differences between linear and nonlinear waves and the special features of solitons. Solve linear waves equations and understand the concept of a dispersion relation. Construct similarity solutions. use the inverse scattering transform and to construct analytical solutions.

General Competences

Search for, analysis and synthesis of data and information, with the use of the necessary technology. Adapting to new situations. Decision-making.

Syllabus

The linear wave theory, Burgers' equation, the Korteweg-de Vries (KdV) equation, travelling waves and the scattering problem for the KdV equation, the inverse scattering transform and solitons, the nonlinear Schrödinger equation, applications to water waves and optics.

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 33 Course total 150
Student Performance Evaluation	Weekly homework Final project Final exam

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Solitons: an Introduction, P. G. Drazin and R. S. Johnson, Cambridge University Press, 1989.
• Γ. Δ. Ακρίβης και Ν .Δ. Αλικάκος, Μερικές Διαφορικές Εξισώσεις, Σύγχρονη Εκδοτική, 2012.
• Εφαρμοσμένα Μαθηματικά, D. J. Logan, Πανεπιστημιακές Εκδόσεις Κρήτης, 2010.

Efficient Algorithms (MAE748)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE748
Semester	7
Course Title	Efficient Algorithms
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 course is introducing advanced algorithmic concepts and techniques. Several optimization problems are examined and solved using algorithmic techniques. Upon a successful completion of the course, the student will be able to: apply advanced algorithmic techniques and methods to solve advanced optimization problems, understand and apply advanced algorithmic analysis methods to study the efficiency of algorithms, analyze and compare the effectiveness of different algorithmic methods, combine advanced algorithmic techniques to solve new problems.

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

Basics: Algorithm analysis (correctness, time and space complexity), asymptotic analysis (worst and average care), recursive algorithms (Strassen’s algorithm for the matrix multiplication problem), lower bounds (comparison-based sorting, the convex-hull problem). Amortized analysis: The accounting, aggregate and potential methods. Minimum spanning trees: The greedy algorithms by Tarjan, Prin and Kruskal. Minimum cuts: The algorithm by Stoer and Wagner. Maximum flows: Basis terminology (flow network, augmenting path, residual network) the max-flow min-cut theorem, the algorithms by Ford και Fulkerson, Edmonds και Karp, and Dinitz. Planar graphs: Basic terns, Euler’s formula, Kurantowski Theorem, the 5-color theorem, drawings of planar graphs: the algorithm by de Fraysseix, Pach and Pollack, the crossing Lemma. Approximation algorithms: simple algorithms of constant approximation factor, Christofides’ algorithm for the traveling salesman problem, approximation schemes for knapsack and bin packing. Randomized algorithms: simple randomized algorithms for verifying polynomial identities and 2-SAT, random walks.

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 33 Course total 150
Student Performance Evaluation	Written final exam. Two exercises.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Convex Analysis (MAE753)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑE753
Semester	7
Course Title	Convex Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 an introduction to convex analysis at undergraduate level. It is desired for students to understand convex sets with respect to some of their qualitative (from a geometric/combinatorial point of view) and quantitative (e.g. volume, surface area) properties together with the study of the corresponding convex functions.
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. Polytopes. Gauge functions and support functions. The Caratheodory. Radon's and Helly's theorems. Minkowski's First theorem. The Brunn-Minkowski inequality. Mixed volumes. Inequalities of isoperimetric type (e.g. the classical isoperimetric inequality and the Blaschke-Santalo inequality). F. John's Theorem. The reverse isoperimetric inequality.

Teaching and Learning Methods - Evaluation

Delivery	Lectures/ Class presentations
Use of Information and Communications Technology	Use of the platform “E-course” of the University of Ioannina
Teaching Methods	Activity Semester Workload Lectures/Presentations 39 Assignments/Essays 33 Individual study 78 Course total 150
Student Performance Evaluation	Students' evaluation by the following: Class presentation - Essays - Assignments Final Written Examination 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

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• R. J. Gardner, Geometric tomography. Second edition. Cambridge University Press, 2006.
• R. Tyrel Rockafellar, Convex Analysis. Princeton University Press, 1970.
• R. Schneider, Convex bodies: the Brunn-Minkowski theory. Second expanded edition. Cambridge University Press, 2014.
• A. C. Thompson, Minkowski Geometry. Cambridge University Press, 1996.
• R. Webster, Convexity. Oxford University Press, 1970.

Seminar in Analysis II (MAE754)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ754
Semester	7
Course Title	Seminar in Analysis II
Independent Teaching Activities	Lectures (Weekly Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	None. However it is desirable to have a strong knowledge of basic notions of differential equations.
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	No
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The ability to write a complete report on a scientific subject. The ability to present this report. Working in teams. The report can be, but not required to be, original. Further details can be determined by the teaching professor.
General Competences	Αnalyse and combine data and information using various technologies. Working independently and in groups. Free, creative, analytic, and conclusive thinking. Decision making. Further details can be determined by the teaching professor.

Syllabus

In depth study in a scientific subject related to mathematical analysis. Further details can be determined by the teaching professor.

Teaching and Learning Methods - Evaluation

Delivery	Details will be determined by the teaching professor. Methods include presentations contacted by the students.
Use of Information and Communications Technology	Details will be determined by the teaching professor.
Teaching Methods	Activity Semester Workload Study in class 39 Other activities determined by the teaching professor 111 Course total 150
Student Performance Evaluation	There is no final exam. Each student must write a report on a specific subject. Each student must present the report publically. Students may miss up to 3 lectures. Other means of evaluation can be determined by the teaching professor.

Attached Bibliography

Bibliography is suggested by the teaching professor, depending on the subject under study.

Seminar in Geometry (MAE761)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ761
Semester	7
Course Title	Seminar in Geometry
Independent Teaching Activities	Lectures (Weekly Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	None. However it is desirable to have a strong knowledge on basic aspects of multivariate real analysis.
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	No
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The ability to write a complete report on a scientific subject in Geometry. The ability to present this report. Working in teams and carrying out projects. The report can be, but not required to be, original. Further details can be determined by the teaching professor.
General Competences	Αnalyse and combine data and information using various technologies. Working independently and in groups. Free, creative, analytic, and conclusive thinking. Decision making. Further details can be determined by the teaching professor.

Syllabus

In depth study in a scientific subject related to geometric topology. Further details can be determined by the teaching professor.

Teaching and Learning Methods - Evaluation

Delivery	Details will be determined by the teaching professor.
Use of Information and Communications Technology	Details will be determined by the teaching professor.
Teaching Methods	Activity Semester Workload Study in class 39 Self Study 78 Projects 33 Course total 150
Student Performance Evaluation	There is no final exam. Each student must write a report on a specific subject. Each student must present the report publically. Students may miss up to 3 lectures. Other means of evaluation can be determined by the teaching professor.

Attached Bibliography

Bibliography is suggested by the teaching professor, depending on the subject under study.

Astronomy (MAE801)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE801
Semester	8
Course Title	Astronomy
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 introduces students to the basic principles of astronomy. Upon successful completion of this course students should be able to: know the physical parameters related to the structure, evolution, and final stages of stars. describe the most important features of the Sun and its activity. know the most important features of the members of our planetary system. ecognize the structure of the Milky Way Galaxy and other galaxies. present the up‐to‐date views about the structure and evolution of the Universe.

General Competences

Search for, analysis and synthesis of data and information, with the use of the necessary technology Working independently Respect for the natural environment Production of free, creative and inductive thinking

Syllabus

Mechanisms of emission and absorption of radiation. Radiative transfer in stellar atmospheres. Stellar magnitudes and distances. Stellar spectra and classification, Hertzsprung‐Russell diagram. Internal structure, formation and evolution of stars. Final stages of stars: white dwarfs, neutron stars and black holes. The Sun and the solar system. Variable and peculiar stars. Stellar groups and clusters. Interstellar matter. The Milky Way Galaxy. Other galaxies. Cosmology.

Teaching and Learning Methods - Evaluation

Delivery	Face to face teaching
Use of Information and Communications Technology	The Moodle e‐learning platform is used for the delivery of lecture notes and exercises to the students.
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Study & analysis of bibliography 90 Non-directed study 18 Examination 3 Course total 150
Student Performance Evaluation	Written examination at the end of semester.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• "Αstrophysics, volume Ι", F. Shu, Crete University Press, ISBN: 978-960-7309-16-7 (in Greek).
• "Αstrophysics, volume ΙI", F. Shu, Crete University Press, ISBN: 978-960-7309-17-4 (in Greek).

Meteorology (MAE802)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE802
Semester	8
Course Title	Meteorology
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 to give students the opportunity to be familiar with the basic principles of Meteorology and realize if they are interested in working, studying or doing research on this scientific field in the future. Specifically, after the successful completion of the course, the students will be able to: Explain the definitions and the quantitative and qualitative characteristics of the various meteorological parameters. Describe and explain the various meteorological phenomena. Describe and explain the main measurement techniques in Meteorology and the meteorological instruments.

General Competences

Search for, analysis and synthesis of data and information, with the use of the necessary technology Working independently Respect for the natural environment Production of free, creative and inductive thinking

Syllabus

Weather and climate. Composition and vertical structure of the atmosphere. Solar radiation and mechanisms of heat transfer in the atmosphere. Air temperature. Atmospheric pressure. Wind. Large-scale and small-scale circulations in the atmosphere. Atmospheric humidity. Atmospheric stability. Clouds, fog, dew and frost. Precipitation (rain, snow, etc.). Fronts. Atmospheric disturbances. Measurement techniques and meteorological instruments. Fundamental elements of weather analysis and forecasting. Educational visit to the Laboratory of Meteorology of the Physics department and the university meteorological station.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	Asynchronous online learning via Moodle is used for providing the lecture slides and the communication with the students.
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Individual study 90 Solving exercises 15 Educational visits 6 Course total 150
Student Performance Evaluation	Written examinations at the end of semester, comprising questions of knowledge and understanding of the course content.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Ahrens CD, Henson Ρ. 2018: Meteorology Today: An Introduction to Weather, Climate and the Environment 12th Edition, Cengage Learning.

Operator Theory (MAE811)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE811
Semester	8
Course Title	Operator Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 goal of the course is the study of inner product and Hilbert spaces (which in the case of finite dimensional spaces are the well-known Euclidean spaces) and the study of bounded, but also of non-bounded, linear maps (linear operators) between them. These operators appear in many branches of theoretical and applied mathematics. For example, they appear in Differential and Integral equations, in Fourier analysis, in quantum mechanics and in quantum information theory. The aim is to transform these operators (where it is possible) into diagonal operators with respect to appropriate "bases". Classes of operators will be studied for which this result is achieved.

General Competences

Αnalyse and combine data and information using various technologies. Working independently and in groups. Free, creative, analytic, and conclusive thinking. Decision making.

Syllabus

Spaces with inner product, Hilbert spaces, basic properties. Orthonormal sets and orthonormal bases in Hilbert spaces. Bounded operators, adjoint operators, orthogonal projections. Finite-order operators, compact operators, Fredholm's Alternative. Operator diagonalization, the spectral theorem for compact normal and in particular self-adjoint operators. Unbounded linear operators.

Teaching and Learning Methods - Evaluation

Delivery	Teaching on the blackboard from the teacher
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Individual study 78 Solving exercises-homework 33 Course total 150
Student Performance Evaluation	Exams in the end of the semester (mandatory), intermediate exams (optional), assignments of exercises during the semester (optional).

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

1. ---

Qualitative Theory of Partial Differential Equations (MAE815)

Qualitative Theory of Partial Differential Equations (MAE815)

Difference Equations - Discrete Models (MAE816)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE816
Semester	8
Course Title	Difference Equations - Discrete Models
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special 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

Remembering: The concept of the Difference Operator, the Summation Operator and the Shift Operator. The concept of Binomial Coefficient and the Gamma Function. The concept of the Generating Function. The concept of the Difference Equation. The concept of the z-Transformation. The concepts of the Stable Fixed Point and the Asymptotically Stable Fixed Point. The concepts of Liapunov Function and Strictly Liapunov Function. The concept of sensitive dependence on initial conditions. The concept of asymptotic relation between functions. The concepts of "O-big" and "O-small". The concept of the homogeneous linear Poincare-type equation. The concept of the boundary value problem for non-linear equations. The concept of Partial Difference Equations. Comprehension: Basic properties of the Difference Operator, the Summation Operator and the Shift Operator. Calculation of indefinite sums. Solving certain types of linear difference equations. Finding fundamental sets of solutions for linear difference equations. Using the Casorati determinant in order to solve linear difference equations. Using Generating Functions and z-Transformations in order to solve difference equations. Linearisation of non-linear difference equations. Studying the stability of the solutions of difference equations and the Floquet Theory. Studying the stability of non-linear systems of difference equations and chaotic behaviour. Asymptotic approximation of sums. Green Functions of boundary value problems for difference equations. Oscillation of solutions for difference equations. Studying the Sturm-Liouville problem. Studying boundary value problems for non-linear difference equations. Studying partial difference equations. Applying: Studying economy-related real world problems. Studying the growth or the decline of populations. Studying physics-related real world problems. Studying probabilities-related real world problems. Studying epidemiology-related real world problems. Evaluating: Teaching undergraduate and graduate courses.

General Competences

Creative, analytical and inductive thinking. Required for the creation of new scientific ideas. Working independently. Working in groups. Decision making.

Syllabus

The Difference Calculus, Linear difference equations, Stability theory, Asymptotic methods, The Sturm-Liouville problem, Boundary value problems for non-linear difference equations, Partial difference equations.

Teaching and Learning Methods - Evaluation

Delivery	Lectures in class. Learning Management System (e.g.: Moodle).
Use of Information and Communications Technology	- Use of Learning Management System (e.g.: Moodle), combined with File Sharing and Communication Platform (e.g.: NextCloud) for distributing teaching material, submission of assignments, course announcements, gradebook keeping for all students evaluation procedures, communicating with students. Use of Web Appointment Scheduling System (e.g.: Easy!Appointments) for organising office appointments. Use of Google services for submitting anonymous evaluations regarding the teacher.
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Preparation of assignments and interactive teaching 33 Course total 150
Student Performance Evaluation	Language of evaluation: Greek and English. Methods of evaluation: Weekly written assignments. Few number of tests during the semester. Based on their grades in the aforementioned weekly assignments and tests, limited number of students can participate in exams towards the end of the semester, before the beginning of the exams period. In any case, all students can participate in written exams at the end of the semester, during the exams period. 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

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Special Topics in Geometry (MAE822)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE822
Semester	8
Course Title	Special Topics in Geometry
Independent Teaching Activities	Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, 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

This course introduces the notion of differential forms. The aim of the course is to prove Stokes theorem for manifolds with boundary and to provide applications in differential geometry as well as in other areas of mathematics. The course requires tools from linear algebra, calculus of several variables, topology and elementary differential geometry. On completion of the course the student should be familiar with differential forms and the meaning of Stokes theorem.

General Competences

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

Syllabus

Differential forms in Euclidean space, line integrals, differentiable manifolds (with or without boundary), integration of differential forms on manifolds, theorem of Stokes and applications, Poincare lemma, differential geometry of surfaces, structure equations.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which includes resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• M. do Carmo, Διαφορικές Μορφές, Θεωρία και Εφαρμογές, Prentice-Hall, Πανεπιστημιακές Εκδόσεις Κρήτης, 2010.

Riemannian Geometry (MAE825)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE825
Semester	8
Course Title	Riemannian Geometry
Independent Teaching Activities	Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, 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

The main task is to present the fundamental concepts of Riemannian geometry, i.e., the concepts of curvatures and differential form on manifolds with boundary. Moreover, we will introduce the notions of Riemannian submanifold and will investigate the Gauss-Codazzi-Ricci equations. The lecture will be completed with the presentation of the sphere theorem, a deep and important result that connects geometry with topology. On the completion of the course we expect that the student fully understand the main theorems that were presented during the lectures.

General Competences

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

Syllabus

Riemannian metrics, curvature operator, Schur's theorem, differential forms, integration on manifolds, Stokes' theorem, Riemannian submanifolds, sphere theorem.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Weekly exercises and homeworks, presentations, final written exam in Greek (in case of Erasmus students in English) which includes resolving application problems.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• M. do Carmo, Riemannian Geometry, Birkhaüser Boston, Inc., Boston, MA, 1992.
• J. Eschenburg, Comparison Theorems in Riemannian Geometry, Universität Augsburg, 1994.
• J. Jost, Riemannian Geometry and Geometric Analysis, Universitext, Springer, 2017.
• J. Lee, Riemannian manifolds: An Introduction to Curvature, Vol. 176, Springer, 1997.
• P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171, Springer, 2016.
• Δ. Κουτρουφιώτης, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων, 1994.

Topological Matrix Groups (MAE826)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE826
Semester	8
Course Title	Topological Matrix Groups
Independent Teaching Activities	Interactive, Presentations (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background, skills development
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, 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

The aim of the course is to provide an introduction to Lie theory through matrix groups. The main subject of study is the closed subgroups of the general linear group. Our study is extended from real to complex and quaternion numbers. The corresponding linear groups are in fact topological groups and an introduction of basic properties of topological group is also provided. The Lie algebra of a matrix group is defined. The special orthogonal, unitary and symplectic groups provide important example of Lie algebras. Lie algebras are studied using the exponential map. Finally Lie groups are defined.

General Competences

Study particular characteristics of group theory in topology and geometry Independent and team work Working in an interdisciplinary.

Syllabus

• General linear groups
• Real and Complex algebras, Quaternions. Matrix algebras
• Inner product, orthogonal, unitary and symplectic groups
• Homomorphisms
• Differential curves, tangent vectors. Dimension of a matrix group
• Differential homomorphisms
• Expontential and logarithmic funcions. Lie algebras
• Special orthogonal and symplectic groups
• Topological groups, manifolds
• Maximal tori
• Differential manifolds, Lie groups.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face, Distance learning
Use of Information and Communications Technology	Communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Working hours in class 8 Project 30 Assignments 33 Final exam 41 Course total 150
Student Performance Evaluation	Written Examination, Oral Presentation, written assignments.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• J. F. Adams, Lectures on Lie groups, University of Chicago Press, 1969.
• M. L. Curtis, Matrix Groups, Springer-Verlag, 1979.
• R. Howe. Very basic Lie theory, American math. monthly,90, 1983.

Statistical Data Analysis (MAE832)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ832
Semester	8
Course Title	Statistical Data Analysis
Independent Teaching Activities	Lectures-Laboratory (Weekly Teaching Hours: 3, Credits: 6)
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	In this course, various statistical methodologies are applied with the help of the computer and the use of statistical programs (SPSS, JASP, R). Emphasis is placed on choosing the appropriate statistical methodology and examining whether the assumptions of its application are met.
General Competences	Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism.

Syllabus

In this course, various statistical methodologies are applied with the help of the computer and the use of statistical programs (SPSS, JASP, R).

Upon completion of the course, the student will be able to:

1. enter data into the computer
2. conduct descriptive statistical analysis, that is to summarize the available data,
3. conduct basic data analyzes (testing hypotheses concerning the mean of a population, the means of two populations with dependent and independent samples, one way analysis of variance etc.,)
4. fit simple-multiple linear regression models, binomial logistic regression models, checking whether the assumptions of their application are violated
5. apply basic methodologies of multidimensional analysis (clustering, factor analysis)
6. present the results of the above analyzes (reference report).

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-Homeworks 33 Course total 150
Student Performance Evaluation	Written final exam in Greek (in case of Erasmus students in English language) which includes the analysis of both real and educational datasets. During the semester, compulsory, usually individual, assignments are given.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Andy Field. Η Διερεύνηση της Στατιστικής με τη χρήση του SPSS της IBM. Εκδόσεις Προπομπός
• Julie Pallant. SPSS Οδηγός ανάλυσης δεδομένων με το ΙΒΜ SPSS. Εκδόσεις Κλειδάριθμος ΕΠΕ.
• Δ. Φουσκάκης. Ανάλυση Δεδομένων με χρήση της R. Εκδόσεις Τσότρας Αν. Αθανάσιος
• Joaquim P. Marques de Sá. Applied Statistics Using SPSS, STATISTICA, MATLAB and R. Springer Berlin Heidelberg Διαθέτης (Εκδότης) HEAL-Link Springer ebooks
• Μαλεφάκη, Σ., Μπατσίδης, Α., & Οικονόμου, Π. (2023). Στατιστική Ανάλυση Δεδομένων [Προπτυχιακό εγχειρίδιο]. Κάλλιπος, Ανοικτές Ακαδημαϊκές Εκδόσεις.

Inventory Control and Production Planning (MAE833)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE833
Semester	8
Course Title	Inventory Control and Production Planning
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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	Upon successful completion of the course, students will: understand the basic principles of control of production and inventory systems understand the performance measures for production and inventory systems and how these measures affect systems cost use appropriate tools to find cost-effective production and inventory policies understand how the variability affects the systems performance.
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. Forecasting (Classical Demand Forecasting Methods, Forecast Accuracy, Machine Learning in Demand Forecasting, Demand Modeling Techniques). Deterministic Inventory Models (Continuous Review: Economic Order Quantity (EOQ) model, EOQ Model with Discounts, EOQ Model with shortages, Economic Production Quantity (EPQ) Model, Periodic review: Wagner-Whitin Algorithm). Stochastic Inventory Models (Continuous Review: Exact (r,Q) policy with Continuous Demand Distribution, Exact (r,Q) policy with Discrete Demand Distribution. Periodic Review with Zero Fixed Costs: Base Stock Policies, Periodic Review with Nonzero Fixed Costs: (s, S) Policies, Policy Optimality). Materials requirement planning (MRP)

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	eCourse
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 78 Fieldwork (3-4 set of homework) 33 Course total 150
Student Performance Evaluation	LANGUAGE OF EVALUATION: Greek METHODS OF EVALUATION: Final exam (100%)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

1. Ιωάννου, Γ., Διοίκηση Παραγωγής και Υπηρεσιών, Εκδόσεις Α. Σταμούλης, Αθήνα-Πειραιάς, 2005.
2. Ξανθόπουλος Αλ., Κουλουριώτης Δημ., Διοίκηση παραγωγής και επιχειρησιακών λειτουργιών: σχεδιασμός, προγραμματισμός και έλεγχος σε συστήματα παραγωγής και υπηρεσιών, Εκδόσεις Τζιολα, 2019.
3. Muckstadt, J. A., A. Sapra. Principles of Inventory Management When You Are Down to Four, Order More. Springer, 2010
4. Nahmias, S., Production and Operations Analysis, McGraw-Hill: Series in Operations and Decision Sciences, 2009.
5. Shenoy D. and R. Rosas. Problems & Solutions in Inventory Management. Springer, 2018
6. Shim, J.K. and Siegel, J.G., Διοίκηση Εκμετάλλευσης (μεταφρ.), Εκδόσεις Κλειδάριθμος: Σειρά Οικονομία και Επιχείρηση, Αθήνα, 2002.
7. Silver, E. A., D. F. Pike, and R. Peterson. Inventory Management and Production Planning and Scheduling, 3rd ed. Hoboken, NJ: Wiley, 1998.
8. Snyder, L. V. and Z-J. M. Shen. Fundamentals of Supply Chain Theory, 2rd ed. John Wiley & Sons, Inc., 2019
9. Tersine R. J. Διαχείριση υλικών και συστήματα αποθεμάτων (μεταφρ.), Εκδόσεις Παπαζηση ΑΕΒΕ, 1984
10. Zipkin, P. Foundations of Inventory Management. McGraw-Hill/Irwin. 2000.
11. Tersine R. J. Διαχείριση υλικών και συστήματα αποθεμάτων (μεταφρ.), Εκδόσεις Παπαζηση ΑΕΒΕ, 1984
12. [Περιοδικό / Journal] Production and Operations Management
13. [Περιοδικό / Journal] Production Planning and Control.

Non Parametric Statistics - Categorical Data Analysis (MAE835)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ835
Semester	8
Course Title	Non Parametric Statistics- Categorical Data Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 aim of this course is to introduce students to the methods of Non parametric techniques (goodness-of-fit tests, ranks etc) as well as their application to real practical problems. At the end of the course the student should have understood the basic methods of Non-Parametric Statistics and Categorical Data, knowing when to adopt and how to apply them for analyzing data.
General Competences	Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism.

Syllabus

Empirical distribution function, Goodness of fit tests: Kolmogorov-Smirnov test, Chi-square, Runs test, Sign tests, Wilcoxon - Mann - Whitney test, Kruskal - Wallis test. Correlation coefficients. Categorical Variables. Statistical inference for binomial and multinomial parameters, Contingency Tables, Comparing two proportions, Testing: independence, Symmetry, Homogeneity. 2 x 2 Tables (Exact Fisher's test, McNemar's test). Applications. Loglinear models.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures (13X3) 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English).

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Agresti, A. (2007). An Introduction to Categorical Data Analysis. 2 ed. ISBN: 978- 0-470-38800-# Wiley
• Conover, W. J. (1999). Practical Nonparametric Statistics. 3 ed. ISBN: 978-0-471- 16068-# John Wiley & Sons
• Ζωγράφος, Κ. (2009). Κατηγορικά Δεδομένα. Πανεπιστήμιο Ιωαννίνων.
• Μπατσίδης, Α. (2010). Εισαγωγή στη Μη Παραμετρική Στατιστική. Πανεπιστήμιο Ιωαννίνων

Computational Statistics (MAE836)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ836
Semester	8
Course Title	Computational Statistics
Independent Teaching Activities	Lectures-Laboratory (Weekly Teaching Hours: 3, Credits: 6)
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	Students completing this course should be able to: Apply the most common methods of computational statistics generate random numbers from discrete and continuous distributions use R and other statistical software to perform statistical analysis use different methods to solve an optimization problem.
General Competences	Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism.

Syllabus

Using R the following topics will be discussed: Generation of random numbers from discrete and continuous distributions. Monte Carlo integration. Using simulation techniques to visualize classical results of statistical inference via simulated data (asymptotic normality of mean, power of a test etc). Density Estimation and Applications (Kernel density estimation). Methods of Resampling ς (Jackknife και Bootstrap). Numerical maximization techniques (Newton-Raphson, Fisher scoring, expectation-maximization [EM]).

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which concentrates on the solution of problems which are motivated by the main themes of the course.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Davison, A. C., Hinkley, D. V., Bootstrap methods and their application. Cambridge University Press 1997.
• Rizzo, M. L., Statistical computing with R. Chapman & Hall/CRC 2007.
• Robert, C. P., Casella, G., Introducing Monte Carlo methods with R. Springer Verlag 2009

Special Topics in Statistics (MAE837)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ837
Semester	8
Course Title	Special Topics in Statistics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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	Students will become familiar with the themes in question and develop knowledge of statistical methods, and will also learn how the methodology becomes relevant in certain application areas. Students will learn a specialized field of statistics not covered by any ordinary course.
General Competences	Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism.

Syllabus

The precise contents of this course may vary from occasion to occasion, but will consist of selected themes of contemporary research interest in statistics 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.
For the next academic year the syllabus of the course is the following: Multivariate distributions: basic properties. Multivariate normal distribution: properties and estimation. Brief review of multivariate methods of statistical analysis: Principal Components, Factor Analysis, MANOVA, Discriminant Analysis.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which concentrates on the solution of problems which are motivated by the main themes of the course.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Σημειώσεις διδάσκοντα
• Everitt, B., Hothorn, T. (2011) An introduction to Applied Multivariate Analysis with R. Springer.
• Hastie, Tibshirani and Friedman (2009) Elements of Statistical Learning, 2nd edition, Springer.
• James, Witten, Hastie and Tibshirani (2011) Introduction to Statistical Learning with applications in R. Springer.
• B.S. Everitt, S. Landau, M. Leese and D. Stahl (2011) Cluster Analysis, 5th Edition, Wiley.

Special Topics in Probability (MAE838)

Special Topics in Probability (MAE838)

Seminar in Operational Research (MAE839)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ839
Semester	8
Course Title	Seminar in Operational Research: Markov Decision Processes and Reinforcement Learning
Independent Teaching Activities	Lectures (Weekly Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	It is desirable to have an elementary knowledge of probability theory, Markov chains, and linear/dynamic programming.
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	No
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The theory of Markov decision processes (MDPs) - also known under the names sequential decision theory, stochastic control or stochastic dynamic programming - studies sequential optimisation of stochastic systems by controlling their transition mechanism over time. In particular, it provides solution methodologies for a wide range of problems concerning sequential decisions in a random environment, statistically modeled by a finite-state Markov chain. The optimal strategy is calculated by appropriate algorithms, which are derived and illustrated in the first part of the course. MDPs have applications in many areas including revenue management (e.g., hotel, airline, and rental car pricing), control of queues, financial engineering, telecommunications, manufacturing, and economics. MDPs provides the mathematical foundation of Reinforcement Learning (RL). RL is one of the most important and emerging categories of Machine Learning, due to the great flexibility of its algorithms in managing large state spaces in problems modeled as MDPs. The aim of the second part of the course is to present the basic principles of Reinforcement Learning, emphasizing both the necessary mathematical framework in which it is structured, and the algorithms, many of which are installed in the R and Matlab programs, for their better understanding.

General Competences

The course aims to enable students to: Become familiar with the general theory and techniques related to MDPs. Become familiar with the basic algorithmic methods for MDPs and become familiar with the reinforcement learning environment. At the end of the course, the student will be able to: evaluate well-known theorems in the field of MDPs, apply algorithmic methods for MDPs to real-world examples using R, Matlab software. implement common RL algorithms in code.

Syllabus

MDPs in discrete time on a finite time horizon, MDPs in discrete time on an infinite time horizon. Properties of the Bellman equation, contraction and monotonicity, policy improvement algorithms, gradient descent, mirror descent and stochastic gradient descent. Basic principles of reinforcement learning, introduction to a simplified subclass of Reinforcement Learning problems also known as Multi-Armed Bandits. Reinforcement learning methods based on successive approximation algorithms (value iteration): Q-learning based on a single trajectory, with and without function approximation, offline and online versions. Reinforcement learning methods based on policy improvement algorithms (policy iteration): Policy gradient, natural policy gradient.

Teaching and Learning Methods - Evaluation

Delivery	Details will be determined by the teaching professor. Methods include presentations contacted by the students.
Use of Information and Communications Technology	Details will be determined by the teaching professor.
Teaching Methods	Activity Semester Workload Study in class 39 Other activities determined by the teaching professor 111 Course total 150
Student Performance Evaluation	Take home problems plus a presentation. You work on the problems in groups of size 2. Other means of evaluation can be determined by the teaching professor.

Attached Bibliography

Bibliography is suggested by the teaching professor, depending on the subject under study.

Parallel Algorithms and Systems (MAE840)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE840
Semester	8
Course Title	Parallel Algorithms and Systems
Independent Teaching Activities	Lectures-Laboratory (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	Introduction to programming, Introduction to Computers, Database Systems and Web applications development
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 knowledge acquisition of: Parallel algorithmic methods, multitasking programming, thread programming, resources contention/congestion and contention/congestion avoidance mechanisms Understanding of the basic functional parts of a parallel and a distributed system. Understanding of the basic concepts and techniques / programming, communication, and transparency techniques used in both parallel and distributed systems. Programming parallel tasks using parallel programming libraries such as OpenMP and distributed programming tools such as MPI. Parallel algorithms, Parallel architectures, Parallel algorithm development, Parallel Selection, Parallel Merge, Parallel Classification, Parallel Search, Parallel Algorithms of Computational Geometry. Parallel iterative methods for solving Linear problems. Parallel and Distributed Systems and Architectures. Performance of Parallel and Distributed Systems and Applications. Threading / multitasking and programming of parallel and distributed algorithms using OpenMP and MPI.

General Competences

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

Syllabus

1. Historical review of parallel and distributed processing.
2. Von Neumann model. Flynn categorization. Tubing. Multiprocessors, Multi-computers.
3. Distributed and Shared Memory Systems. Memory architectures for single and non-unified access time.
4. Performance calculations and metrics. System scalability, partitioning and optimization. Parallel computer interface networks.
5. Law of Grosch, of Amdahl, of Gustafson Barsis. Design of parallel applications.
6. Program parallelization - MPI. Synchronization. Dependency charts, shared resources and racing conditions. Scheduling. Shared Memory Affinity. MESI. Parallel Processing using parallella FPGA cores.
7. Models and process communication mechanisms. Vector Processing. Arrays and computational grid. Examples of application parallelization. Synchronization issues

Course laboratory part

1. Introductory programming concepts using gcc. Pointers, classes, dynamic structures. Creating processes in Linux, separating user-space and kernel-space concepts, parenting processes and parent-child relationships, Process Management.
2. Containers, Templates, STL (C++ standard templates library).
3. Introduction to Boost and advanced C ++ aspects.
4. Introduction to C ++ Armadilo
5. Process intercommunication. Static memory areas, pipelines, shared memory areas, process signalling.
6. Threads creation and thread management. shared thread memory areas, critical areas, producer-consumer model, threads signalling.
7. Thread Management and Synchronization, critical areas protection using mutex locks and semaphores. Presentation of conditional execution threads and sync barriers.
8. Introduction to MPI, MPI settings, MPI key features presentation, preliminary MPI programs.
9. Presentation of basic modern methods of sending and receiving messages in MPI. Presentation of asynchronous upload methods. Examples.
10. Using Gather-Scatter-Reduce-Broadcast Collective Methods and Examples.
11. Basic structures for organizing distributed programs. Examples of distributed calculations. Advanced data types using MPI. Creating # Complex Data Structures with MPI And Sending Data Structure Messages.
12. Parallel programming OpenMP and Epiphany-SDK, BSP.

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 33 Course total 150
Student Performance Evaluation	Using new ICT and metrics of the asynchronous e-learning platform (30%) Examination of laboratory exercises (20%) Semester written examination (50%)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Parallel Scientific Computing in C++ and MPI: A Seamless Approach to Parallel Algorithms and their Implementation, G.M. Karniadakis and R.M. Kirby, 2003, Cambridge University press, ISBN: 0-521-81754-4
• Using OpenMP, Portable Shared Memory Parallel Programming., B. Chapman, G. Jost and R. Pas, 2008, MIT press, ISBN: 9780262533027
• Learning Boost C++ libraries, A. Mukherjee, 2015, PACKT, ISBN:978-1-78355-121-7
• Boost C++ Application Development Cookbook - Second Edition: Recipes to simplify your application development, 2 Edition, A. Polukhin, 2017, PACKT, ISBN:978-1-78728-224-7
• C++17 STL Cookbook, J. Galowicz, PACKT,978-1-78712-049-5, 2017

Special Topics in Computer Science (MAE841)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE841
Semester	8
Course Title	Special Topics in Computer Science
Independent Teaching Activities	Lectures, exercises, tutorials (Weekly Teaching Hours: 3, Credits: 6)
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 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-Homeworks 33 Course total 150
Student Performance Evaluation	Final written examination (70%) Exercises / Homework (30%)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Παπαδόπουλος, Α., Μανωλόπουλος, Ι., Τσίχλας, Κ. 20# Εισαγωγή στην Ανάκτηση Πληροφορίας, Αποθετήριο «Κάλλιπος», 2015.
• Παρασκευάς, Μιχαήλ, Ειδικά θέματα εφαρμογών της Κοινωνίας της Πληροφορίας, Αποθετήριο «Κάλλιπος», 20#
• Δημακόπουλος, Β. Εισαγωγή: Παράλληλα Συστήματα και Προγραμματισμός, Αποθετήριο «Κάλλιπος», 2015.

Special Topics in Numerical Analysis (MAE842)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ842
Semester	8
Course Title	Special Topics in Numerical Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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: thoroughly understand problems arising from applications' be aware to analyze the problem and chose the appropriate numerical method for solving it, solve the problem by implementing the 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

Special subjects of Numerical Linear Algebra coming from Applications. Special subjects of Numerical Solution of Differential Equations coming from 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 bibliografy 104 Exercises-Homeworks 33 Course total 150
Student Performance Evaluation	Written examination, Project.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Special Topics in Applied Mathematics (MAE843)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE843
Semester	8
Course Title	Special Topics in Applied Mathematics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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	Face to face
Use of Information and Communications Technology	Use of computer (Mechanics) lab
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Exercises 33 Course total 150
Student Performance Evaluation	Weekly homework Final project Final exam

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Algorithm Engineering (MAE844)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE844
Semester	8
Course Title	Algorithm Engineering
Independent Teaching Activities	Lectures, laboratory exercises, tutorials, quiz (Weekly Teaching Hours: 3, Credits: 6)
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 aims at introducing to students the concepts , techniques, properties, developments and applications of basic and advanced algorithms and data structures. Software development and software libraries that allow to easily develop and evaluate experimentally algorithms. Methodologies related to experimental research of efficient algorithms and data structures. After successfully passing this course the students will be able to: Understand basic algorithmic techniques Analyze complex algorithms Design and develop new algorithmic tools for experimental evaluation

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

• Introduction to algorithm engineering
• Methodology of Algorithm Engineering: motivation, applications, software systems
• System checking
• Software reliability and correctness
• STL and Generalized programming
• Experimental evaluation of algorithms

Teaching and Learning Methods - Evaluation

Delivery	Lectures
Use of Information and Communications Technology	Use of projector and interactive board during lectures. Course website maintenance. Announcements and posting of teaching material (lecture slides and notes, programs). Announcement of assessment marks via the ecourse platform by UOI.
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Team work 33 Course total 150
Student Performance Evaluation	Final written examination (70%) Exercises (30%)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• K. Mehlhorn and S. Naeher, LEDA: A platform for combinatorial and geometric computing, Cambridge University Press, 1999.
• M. Mueller-Hannemanni and S. Schirra, Algorithm Engineering - Bridging the Gap between Algorithm Theory and Practice, Springer 2010.
• C.C. McGeoch, A Guide to Experimental Algorithmics, Cambridge University Press, 2012.
• J. Siek, L.Q. Lee, and A. Lumsdaine, The Boost Graph Library, Addison-Wesley, 2002.
• M.A. Weiss, Data structures and problem solving with C++, 2 Edition, Addison-Wesley, 2000.

Introduction to Natural Languages Processing (MAE845)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE845
Semester	8
Course Title	Introduction to Natural Language Processing
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 goal of this course is the deeper understanding of Natural Language Processing (NLP). During the course a detailed examination of the following topics are done: A historical retrospection of Language Technology evolution The goal of NLP and its Applications The NLP levels. Language Processors such as recognition machines, transducers, parsers and generators The language as a rule based system. Language Understanding as process NLP Resources for parsing, such as Data Base, Knowledge Base, Data Structure, Algorithms and Expert Systems Fundamental parsing strategies concerning context free grammars. Fundamental Methods of Computational Morphology, Computational Semantics and NLP. Implementations-Applications After completing the course the student can handle: theoretical documentation of problems solving exercises tracking applications which related to NLP different topics.

General Competences

Handle new problems Decision making Implementation- Consolidation

Syllabus

• A historical retrospection of Language Technology evolution
• The goal of NLP and its Applications
• The NLP levels. Language Processors such as recognition machines, transducers, parsers and generators
• The language as a rule based system. Language Understanding as process
• NLP Resources for parsing, such as Data Base, Knowledge Base, Data Structure, Algorithms and Expert Systems
• Fundamental parsing strategies concerning context free grammars.
• Fundamental Methods of Computational Morphology, Computational Semantics and NLP. Implementations-Applications

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	Yes , Use of Natural Language and Mathematical Problems Processing Laboratory
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Exercises 33 Course total 150
Student Performance Evaluation	Final test

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• 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
• Professor's Notes.

Introduction to Expert Systems (MAE846)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE846
Semester	8
Course Title	Introduction to Expert Systems
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	Logic Programming, Data Structure
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 PROLOG. During the course a detailed examination of the following topics are done: Procedural and Declarative Programming Logic Programming a version of Declarative Programming The programming language PROLOG (PROLOG programs syntax, Lists, Operators, Arithmetic, Backtracking control, The negation in PROLOG, Recursive predicates, Data Structure manipulation, PROLOG implementation to searching problems, symbolic processing, natural language understanding and metaprogramming) Logic Programming Theory Logic Programming under restrictions Logic Programming systems implementation technics Parallel Logic Programming Logic Programming for knowledge representation. After completing the course the student can handle: theoretical documentation of problems solving exercises implementations-applications

General Competences

Applications Implementation- Consolidation

Syllabus

• Ιntroduction to Expert Systems
• Main Features of Expert Systems, classic examples
• Knowledge acquisition and verification, knowledge representation, inference and interpretation, consistency and uncertainties.
• Inference techniques
• Rule-based forward chaining Expert Systems
• Rule-based backward chaining Expert Systems
• Rule-based Expert Systems
• Expert Systems tools
• Users Interface
• Machine learning, decision making machines, Expert Systems examples.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	Yes , Use of Natural Language and Mathematical Problems Processing Laboratory
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Exercises 33 Course total 150
Student Performance Evaluation	Final test

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Γεώργιος Ι. Δουκίδης, Μάριος Κ. Αγγελίδης, "Έμπειρα συστήματα, τεχνητή νοημοσύνη και LISP", ISBN 960-08-0004-9, ISBN-13 978-960-08-0004-3
• Σπύρος Τζαφέστας, "ΕΜΠΕΙΡΑ ΣΥΣΤΗΜΑΤΑ ΚΑΙ ΕΦΑΡΜΟΓΕΣ", ISBN: - (Κωδικός Βιβλίου στον Εύδοξο: 89871)
• Παναγιωτόπουλος Ιωάννης - Χρήστος Π., "Νέες Μορφές Τεχνολογίας - Γενικευμένα Αυτόματα Συστήματα - Έμπειρα Συστήματα Turbo Prolog"

Fluid Mechanics (MAE847)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ847
Semester	8
Course Title	Fluid Mechanics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 is an introduction to the basic concepts of Fluid Mechanics. Upon successful completion of the course, the student will be able to: apply basic concepts of Fluid Mechanics understand and apply advanced analytical and approximate techniques to fluid mechanics problems critically analyze and compare the effectiveness of methods and deepen their further understanding combine advanced techniques to solve new problems in Fluid Mechanics field

General Competences

The course aims to enable the student to be able analyze and synthesize basic knowledge of Fluid Mechanics and Applied Mathematics. Search for, analysis and synthesis of data and information, with the use of the necessary technology Adaptation to new situations Autonomous work Decision making Work in an interdisciplinary environment

Syllabus

• Physical properties of fluids
• Static of fluids
• Kinematics of fluids
• Conservation of mass - continuity equation and Stream function
• Differential equations of motion for ideal fluids - Euler equations, Differential equations of motion for viscous fluids - Navier-Stokes equations
• Applications of Fluid Mechanics.

Teaching and Learning Methods - Evaluation

Delivery	Provision of study material through the ecourse Communication with students through e-mails, and the ecourse and MS Teams platforms
Use of Information and Communications Technology	Use of computer (Mechanics) lab
Teaching Methods	Activity Semester Workload Lectures 39 Study of theory 78 Home exercises 33 Course total 150
Student Performance Evaluation	Weekly assignments Final project Written examination at the end of the semester

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• ---

Scientific Computing (MAE848A)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE848A
Semester	8
Course Title	Scientific Computing
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 most scientific disciplines, the integration of computers has defined new directions to perform research and has offered unprecedented potential to solve complicated problems. Combined with theory and experimentation, computational analysis is nowadays considered an integral part of science and research. The main objective of the course is to familiarize the student with computational techniques that find application in the solution of ordinary and partial differential equations. In the context of this laboratory course, the student will gain access to the programming languages Matlab/Octave and Python, which are widely used to perform scientific calculations. Computational methods to be developed and implemented in PCs will significantly increase the skills and prospects of integrating graduates into the modern scientific and work environment. Starting from the mathematical modeling of problems of Mechanics and Applied Mathematics in general, and by synthesizing information from numerical analysis and numerical solution of ordinary and partial differential equations, students will acquire crucial knowledge in solving mathematical problems by computational means. <br\> Specifically, the objectives of the course are: Familiarity with the Matlab/Octave and Python programming languages to implement numerical methods, solve mathematical problems and graphically design the numerical results Apply numerical derivation using the Finite Difference method Analysis of the numerical schemes resulting from the Finite Difference method Solving ordinary differential equations using one-step and multi-step methods Solving parabolic and elliptic Partial Differential Equations with the Finite Difference Method Theoretical analysis of the Finite Element method Solving parabolic and elliptic Partial Differential Equations with the Finite Element method.

General Competences

The course aims to enable the student to: Search, analyze and synthesize data and information, using the available technologies Work autonomously Work in a team Promote free, creative and inductive thinking.

Syllabus

• Initial Value Problems
• Boundary Value Problems
• Finite Difference method
• Equations of Difference
• Shooting methods and Method of undetermined coefficients
• One-step Methods (Euler, Taylor, Runge-Kutta)
• Multi-step Methods (Adams-Bashforth, Adams-Moulton, Predictor-Corrector)
• Finite Element Method (Galerkin).

Teaching and Learning Methods - Evaluation

Delivery	In the laboratory
Use of Information and Communications Technology	Use of scientific computing software packages
Teaching Methods	Activity Semester Workload Lectures 39 Study of bibliography 39 Laboratory exercises 39 Home exercises (project) 33 Course total 150
Student Performance Evaluation	Weekly assignments Final project Written examination at the end of the semester

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Numerical Methods for Ordinary Differential Equations, 2 Edition, G.D. Akrivis, V.A. Dougalis, 2012 (in Greek).
• A Primer on Scientific Programming with Python, H. P. Langtangen, Springer-Verlag Berlin Heidelberg, 5 Edition, 2016.
• Programming for Computations- MATLAB/Octave, S. Linge, H. P. Langtangen, Springer International Publishing, 2016 (in Greek).
• The Mathematical Theory of Finite Element Method, S. C. Brenner, L. R. Scott, Springer-Verlag, New York, 2008.
• Automated Solution of Differential Equations by the Finite Element Method, A. Logg, K.-A. Mardal, G. N. Wells, Springer-Verlag Berlin Heidelberg, 2012.

Calculus of Variations (MAE849)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE849
Semester	8
Course Title	Calculus of Variations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	Classical Mechanics
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

Calculus of Variations deals with optimisation problems where the variables, instead of being finite dimensional as in ordinary calculus, are functions. This course treats the foundations of calculus of variations and gives examples on some (classical and modern) physical applications. After successfully completing the course, the students should be able to: give an account of the foundations of calculus of variations and of its applications in mathematics and physics. describe the brachistochrone problem mathematically and solve it. solve isoperimetric problems of standard type. solve simple initial and boundary value problems by using several variable calculus. formulate maximum principles for various equations.

General Competences

Search for, analysis and synthesis of data and information, with the use of the necessary technology. Adapting to new situations. Decision-making.

Syllabus

The Euler-Lagrange equation. The brachistochrone problem. Minimal surfaces of revolution. The isoperimetric problem. Fermat's principle (geometric optics). Hamilton's principle. The principle of least action. The Euler-Lagrange equation for several independent variables. Applications: Minimal surfaces, vibrating strings and membranes, eigenfunction expansions, Quantum mechanics: the Schrödinger equation, Noether's theorem, Ritz optimization, the maximum principle.

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 33 Course total 150
Student Performance Evaluation	Weekly homework Final project Final exam

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• Calculus of Variations, I. M. Gelfand and S. V. Fomin, Dover Publications, 2000.
• Εφαρμοσμένα Μαθηματικά, D. J. Logan, Πανεπιστημιακές Εκδόσεις Κρήτης, 2010.
• Θεωρητική Μηχανική, Π. Ιωάννου και Θ. Αποστολάτος, ΕΚΠΑ, 2007.

Seminar in Differential Equations (MAE852)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ852
Semester	8
Course Title	Seminar in Differential Equations
Independent Teaching Activities	Lectures (Weekly Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	None. However it is desirable to have a strong knowledge of basic notions of differential equations.
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	No
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The ability to write a complete report on a scientific subject. The ability to present this report. Working in teams. The report can be, but not required to be, original. Further details can be determined by the teaching professor.
General Competences	Αnalyse and combine data and information using various technologies. Working independently and in groups. Free, creative, analytic, and conclusive thinking. Decision making. Further details can be determined by the teaching professor.

Syllabus

In depth study in a scientific subject related to differential equations, ordinary or partial or stochastic. For example, subjects related to existence and uniqueness of solutions, qualitative theory, methods of finding solutions, tools and theory from other fields of mathematics which can be used to study differential equations, and applications. Further details can be determined by the teaching professor.

Teaching and Learning Methods - Evaluation

Delivery	Details will be determined by the teaching professor. Methods include presentations contacted by the students.
Use of Information and Communications Technology	Details will be determined by the teaching professor.
Teaching Methods	Activity Semester Workload Study in class 39 Other activities determined by the teaching professor 111 Course total 150
Student Performance Evaluation	There is no final exam. Each student must write a report on a specific subject. Each student must present the report publically. Students may miss up to 3 lectures. Other means of evaluation can be determined by the teaching professor.

Attached Bibliography

Bibliography is suggested by the teaching professor, depending on the subject under study.

Numerical Solution of Partial Differential Equations (MAE882)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE882
Semester	8
Course Title	Numerical Solution of Partial Differential Equations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
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 the course, students will be able to: describe and apply key numerical methods for the solution of boundary/initial value problems for elliptic and parabolic equations (e.g., Laplace, Heat Equation). be aware of the optimal order of accuracy of key numerical methods as well as the limitations that may be required in the discretization parameters to ensure stability. apply common techniques for analyzing finite difference and finite element methods. implement finite difference and finite element methods (in Octave ἠ Python) to compute the numerical solution of PDEs and calculate their experimental order of convergence.

General Competences

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

Syllabus

• Finite difference approximations to derivatives.
• The two-point boundary value problem. Boundary conditions of type Dirichlet, Neumann, and Robin.
• Finite differences schemes for the two-point boundary value problem. Consistency and stability. The energy method. Order of accuracy and convergence.
• The Finite Element Method (FEM) for the two-point boundary value problem. A priori and a posteriori estimates. Implementation of FEM.
• Finite differences and Finite element methods for the Heat Equation in 1D. Explicit- and implicit Euler, the Crank-Nicolson method. Consistency and stability.
• The finite element method for elliptic and parabolic equations in higher dimensions.

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 e-learning platform. Provision of study materials in Moodle e-learning platform. 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) for the implementation of the numerical algorithms.
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 75 Directed study of Computer-based exercises 6 Non-directed study of Computer-based exercises 30 Course total 150
Student Performance Evaluation	Computer-based exercises with oral examination (Weighting 35%, addressing learning outcomes 2 and 4) Written examination (Weighting 65%, addressing learning outcomes 1-3)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• “Αριθμητική επίλυση μερικών διαφορικών εξισώσεων”, Π. Χατζηπαντελίδης, & Μ. Πλεξουσάκης, Κάλλιπος, (2015). http://hdl.handle.net/11419/665
• “Αριθμητικές Μέθοδοι για Συνήθεις Διαφορικές Εξισώσεις”, Γ. Δ. Ακρίβης, & Β. Α. Δουγαλής., Πανεπιστημιακές Εκδόσεις Κρήτης, Ηράκλειο, 2η έκδοση, 2013.
• “The mathematical theory of finite element methods”, S. C. Brenner & L. R. Scott (Third ed., Vol. 15), Springer, New York, 2008.
• “Partial differential equations with numerical methods”, S. Larsson, & V. Thomée, Springer-Verlag, Berlin, 2009.

Seminar in Algorithmic Optimization (MAE883)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ883
Semester	8
Course Title	Seminar in Algorithmic Optimization
Independent Teaching Activities	Lectures (Weekly Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	None. However it is desirable to have a strong knowledge of basic notions in Data Structures and Analysis of Algorithms.
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	No
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The ability to write a complete report on a scientific subject. The ability to present this report. Working in teams. The report can be, but not required to be, original. Further details can be determined by the teaching professor.
General Competences	Αnalyse and combine data and information using various technologies. Working independently and in groups. Free, creative, analytic, and conclusive thinking. Decision making. Further details can be determined by the teaching professor.

Syllabus

The course deals each time with a modern topic related to the area of algorithmic optimization and focuses on optimization problems with proven efficient complexity as well as on efficient methods for finding approximate solutions. In the context of the seminar, emphasis is given on the solution methods as well as the algorithmic techniques that can be exploited to find optimal solutions. Further details can be determined by the teaching professor.

Teaching and Learning Methods - Evaluation

Delivery	Details will be determined by the teaching professor.
Use of Information and Communications Technology	Details will be determined by the teaching professor.
Teaching Methods	Activity Semester Workload Study in class 39 Self Study 75 Exercises 33 Course total 150
Student Performance Evaluation	There is no final exam. Each student must write a report on a specific subject. Each student must present the report publically. Students may miss up to 3 lectures. Other means of evaluation can be determined by the teaching professor.

Attached Bibliography

Bibliography is suggested by the teaching professor, depending on the subject under study.