Course Title Calculus Ι · Bibliography S.Ntoyas (in Greek) Calculus . Ι, Leader books. M.Spivak...

Course Title Calculus Ι

Course Code Mas 101

Course Type Compulsory

Level Undergraduate

Year / Semester 1st year/2nd semester

Teacher’s Name

ECTS 8 Lectures / week 2 Laboratories / week

1

Course Purpose and Objectives

Mathematically rigorous foundation of real numbers and limits, aiming to understanding the underlying principles beyond the level of calculations

Learning Outcomes Upon successful completion of this course, students are expected to be able to:

• Know and apply the (ε, δ) definitions for the limits of functions and the (ε, n_0) definitions for the limits of sequences of real numbers.

• Identify discontinuous functions defined in dense subsets of R. • Understand the concept of the derivative and be able to use it in applications. • Understand and use the fundamental theorems in Continuity and the ones related

to the Derivatives of Functions. • Find limits of functions, derivatives and limit of sequences.

Prerequisites There are no prerequisites for this course.

Required There are no co-requisites for this course.

Course Content Fundamental properties of real numbers. Sup and Inf of a set and its basic properties. Sequences, its limits, properties of converging sequences. Subsequences. Nested interval principle. Functions and their limits. Sequential approach to limits of functions. Continuity of functions. Intermediate value Theorem and Existence of Extreme values Theorem. Uniform Convergence. Derivatives, basic results. Mean value Theorem and its variations. Continuity and derivative of Inverse function. Graph of a function. L’Hôpital’s Rule.

Teaching Methodology Classroom instruction

Bibliography S.Ntoyas (in Greek) Calculus Ι, Leader books.

M.Spivak : Calculus

Assessment 2 Midterm exams, final exam

Language Greek

Course Title Calculus II

Course Code MAS 102


Level Undergraduate

Year / Semester 2nd year/3rd semester

Teacher’s Name


1


Mathematically rigorous foundation of Riemann integral and its applications,

Main objective being the introduction to view mathematical objects beyond calculations.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Know and apply the definition of the Riemann integral using partitions and upper and lower sums. • Characterize and demonstrate the integrability of bounded and not necessarily continuous functions. • Know and apply the Fundamental Theorems of Integration. • Know the integration techniques and apply them in computing volumes of solids and areas of level surfaces. • Know and use the Taylor formula to approximate polynomial functions and calculate the associated errors. • Examine the convergence of infinite series.

Prerequisites MAS 101 Required There are no co-requisites for this course.

Course Content Partitions, upper and lower sums, Riemann integral on a closed interval. Basic existence theorems of integrals. Computation of volumes and areas. The Fundamental Theorems of Calculus, generalised integrals. Logarithmic and exponential functions. Basic methods of integration, integration by parts, substitution, induction formulas, integration of rational functions. Taylor’s formula, computation of Taylor’s formula for various basic functions. Approximation of smooth functions by polynomials, the irrationality of e. Series, comparison test, Cauchy’s criterion, ratio test, nth root test, integral test, absolutely and conditionally convergentseries, Leibniz’s Theorem for alternating series, Abel’s and Dirichlet’s criteria, products of series.

Teaching Methodology Lectures

Bibliography S. Ntouyas (In Greek): Calculus 2, Leader Books.

M. Spivak: Calculus

Assessment 2 Midterm exams, final exam, homework.

Language Greek

Course Title Linear Algebra Ι

Course Code MAS 121


Level Undergraduate

Year / Semester 1st year / 1st semester

Teacher’s Name


1


Introduction to basic concepts related to vector spaces and linear maps as well as applications of the theory.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Understand the basic notions of the theory, and be able to use them in concrete

problems. • Understand the relation between matrices and systems of linear equations and linear

maps. • Solve linear systems of equations. • Compute bases for the Row space, Column space, and the Kernel of a matrix, as well

as compute the Rank of a matrix. • Compute determinants and to find the inverse of a matrix. • Compute eigenvalues and bases of eigenspaces. • Develop abstract reasoning and become more familiar with writing proofs.



Course Content The algebra of matrices, invertible matrices. Reduced echelon form of a matrix and linear systems of equations. Vector spaces, base, dimension. Linear maps, matrix of a linear map, change of basis matrix, rank of a matrix. Determinants. The set of solutions of a linear system. Eigenvalues, eigenvectors and eigenspaces.

Teaching Methodology Lectures and Exercises (4 hours per week)

Bibliography 1. Notes from the lectures. 2. Δ. Βάρσος κ.α., Μια Εισαγωγή στη Γραμμική Άλγεβρα, τόμος Ι, Εκδ. Σοφία. 3. D. Lay, Linear algebra and its applications, 4th ed., Addison-Wesley. 4. S. Lang, Linear Algebra, 3rd ed., UTM. 5. T.S. Blyth and E.F. Robertson, Basic Linear Algebra, 2nd ed., SUMS.

Assessment Final exam, 2 midterm exams

Language Greek

Course Title Linear Algebra II

Course Code MAS 122


Level Undergraduate

Year / Semester 1st year / 2nd semester

Teacher’s Name


1


Introduction to basic notions and results concerning normal forms of a linear transformation with applications.


• Know the polynomial ring and apply the Euclidean division. • Know the theory of polynomial divisibility and the representation of polynomials in

irreducible factors. • Understand and apply the diagonalization criteria and diagonalize matrices finding

a suitable basis. • Find the Jordan canonical form and a suitable basis. • Apply the Gram-Schmidt orthogonalization method.



Course Content Polynomial Ring. Characteristic polynomial, diagonalization, applications. Cayley-Hamilton theorem, minimal polynomial. Invariant subspaces, generalized eigenspaces. Primary decomposition theorem. Nilpotent endomorphisms, Jordan canonical form. Inner-product spaces, Gram-Schmidt method. Special matrices and their properties.

Teaching Methodology Lectures (4 hours per week)

Bibliography Lecture notes. Εισαγωγή στη Γραμμική Άλγεβρα, Τόμος Β, Νέα Έκδοση, Δ. Βάρσος, Δ. Δεριζιώτης, Μ. Μαλιάκας, Ο. Ταλέλλη, Εκδόσεις Σοφία, 2005. Τ. S. Blyth and E. F. Robertson, Further Linear Algebra.

Assessment Final exam, 2 Midterm exams.

Language Greek

Course Title Basic Mathematics I

Course Code MAS 131


Level Undergraduate

Year / Semester 1st year/ 1st semester

Teacher’s Name


0


The objectives of this course is to equip first-year undergraduate students with basic techniques of mathematical analysis. The course purpose is the understanding of differentiation, integration, elementary ODEs and complex numbers and the use of related techniques to solve mathematical problems.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Differentiate functions of a single variable and compute limits using appropriate

rules. • Apply differentiation to find the graph of a function and solve optimization problems. • Integrate functions of a single variable. • Compute areas of regions in the plane, volumes of solids of revolution, arc lengths

of curves, and areas of surfaces of revolution using integration. • Solve elementary first order ordinary differential equations. • Perform complex number operations, express complex numbers in polar form and

find the roots of complex equations.



Course Content Functions and limits. Differentiation. Applications of differentiation, graphs, optimization problems. Integration (indefinite, definite and improper integrals), techniques of integration. Applications of integration (areas of domains in the plane, volumes of solids, arc lengths of curves and areas of surfaces of revolution). Differential equations. Complex numbers.


Bibliography Thomas and Finney, Calculus with Analytic Geometry, 9th edition, Addison Wesley. / Greek translation: Calculus, Vols. I & II, G. B. Thomas and R.L. Finney, Crete University Press (1997)

Calculus, J. Stewart, Brooks/Cole (1995).

Assessment Final Exam, 2 Midterm exams.

Language Greek

Course Title Basic Mathematics II

Course Code MAS 132


Level Undergraduate


Teacher’s Name

ECTS 7 Lectures / week 2 Laboratories / week 0


The main objective of this course is the introduction of students to basic calculus techniques in two and three-dimensional Euclidean space. The course’s purpose is to provide understanding of differentiation and integration of multivariable scalar functions and vector functions on regions in the plane or three-dimensional space and the use of related techniques to solve mathematical problems.


• Identify and describe regions in the plane and the three-dimensional Euclidean space. • Differentiate and integrate multivariable functions and find their linear approximation,

while at the same time visualize these concepts through the function’s graph. • Parametrize curves and understand the meaning of velocity vector and arc length. • Give a geometric interpretation to linear transforms of the plane and three-dimensional

space as well as to the classical transformations into polar, cylindrical and spherical coordinates. They should also be able to describe the behavior of functions in terms of these coordinates.



Course Content Analytic Geometry in R2: Vectors, inner product, length, distance between points. Equation for a line, tangent, vertical line to a curve. Circles, ellipses, parabolas, hyperbolas. Analytic Geometry in R3: Vectors, algebraic, geometric properties. Inner product, length, distance between points. Equation for a line (parametric-vector, cartesian format), distance of a point to a line. Regions in Euclidean space. Functions: Curves in the plane, regions between curves, curve intersections. Graphs of functions in R3, analytically and implicitly defined. Solids bounded by surfaces and intersections of surfaces. Transformations: Linear transforms, linear independence and geometric interpretation of determinant. Geometric transforms (translation, rotation, reflection, orthogonal transforms). Polar, cylindrical and spherical coordinates and regions defined in these coordinates. Curves: Curve parametrization in R2 and R3. Velocity, acceleration and tangent line. Arc length. Differentiation: Partial derivatives of multivariable functions. Tangent plane and linear approximation. Gradient and directional derivative. Integration: Double integrals over rectangles and general regions of R2.


Bibliography Thomas and Finney, Calculus with Analytic Geometry, 9th edition, Addison Wesley. Marsden and Tromba, Vector Calculus, Freeman. Stewart, Multivariable Calculus.

Assessment Final Exam, 2 Midterms.

Language Greek

Course Title Sets and Algebraic Structures Course Code MAS 133 Course Type Compulsory Level Undergraduate Year / Semester 1st year/1st semester Teacher’s Name Christoforou Cleopatra


http://www.ucy.ac.cy/ucy.ac.cy/en/people/kleopatr


The main objective of this course is to introduce first-year undergraduate students to rigorous mathematical proofs of theorems, and to enable them to produce proofs using mathematically sound arguments. Students who complete this course should be able to understand basic notions of Set Theory, Number Theory and Algebra, basic properties of functions, as well as use mathematical induction and proof by contradiction to prove mathematical statements, and appreciate mathematical rigor.


• Comprehend basic notions of Set Theory and Functions. • Reason and conduct mathematically rigorous arguments via propositional logic to prove

a statement. • Prove theorems and propositions using induction and proof by contradiction. • Use basic notions of Number Theory • Apply the Euclidean algorithm and deduce statements using the Fundamental Theorem

of Arithmetic. • Identify algebraic structures such as groups, subgroups, rings and fields.



Course Content Set Theory: Sets, subsets. Set operations, complement, De Morgan’s laws, power set. Cartesian product. Relations, equivalence relations (equivalence classes modulo m, projective space, rational numbers). Venn diagrams. Elements of propositional logic (quantifiers, negation, truth diagrams). Functions: Image of a set, inverse image. Inverse function. Composition of functions, graphs. Sets of functions. Countable sets, uncountable sets. Diagonal procedure. Reductio ad absurdum and Mathematical Induction. Well Ordering Principle and Principle of Mathematical Induction. Examples from Number Theory and other areas of mathematics for understanding the procedure for proving a statement using these methods. Number Theory: Divisibility. Greatest common factor and least common multiple. Euclidean algorithm. Fundamental Theorem of Arithmetic. Applications to polynomials. Introduction to Algebraic Structures: Binary operations. Closure of operations. Properties of closed operations. Examples (composition of functions, matrix multiplication, inverse, congruence classes). Subgroups, groups (examples from cyclic groups (complex unit roots), symmetric group). The group (Ζn , +) as a quotient. Rings, fields and solving first order equations ax = b.


Bibliography Geoff Smith, Introductory Mathematics: Algebra and Analysis, Springer, 2nd, 1998. I. Kaplansky, Set Theory and Metric Spaces, American Mathematical Society Chelsea Publishing 1972 (Second edition 2001).


Language Greek

Course Title Mathematics with computers

Course Code MAS 191


Level Undergraduate


Teacher’s Name


2


Introduction of students to MATLAB and to programming.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Familiarize themselves with the MATLAB environment and functions. • Be able to programme in MATLAB and solve problems using m-files. • Be able to make good quality graphs in 2 and 3 dimensions and create or read files. • Know the limitations of numerical calculations and the sources of errors. • Solve problems using MATLAB’s symbolic toolbox.



Course Content MATLAB’s environment. MATLAB functions. For, while and if loops. Graphics in two and three dimensions. Programming. Polynomials. Reading from and writing in files. Computer arithmetic and error propagation. Symbolic computing. Special topics and applications (solution of nonlinear algebraic equations and linear systems, eigenvalue problems, numerical integration, ordinary differential equations).

Teaching Methodology Lectures (4 hours per week), Laboratory sessions (2 hours per week)

Bibliography Γ. Γεωργίου και Χρ. Ξενοφώντος, Εισαγωγή στη MATLAB, Καντζιηλάρης, Λευκωσία (2007). C.B. Moler, Numerical Computing with MATLAB, SIAM, 2004. D. Higham and N. Higham, MATLAB Guide (2nd edition), SIAM, 2005.

Assessment Home and Lab Assignments, two mid-exams, final exam.

Language Greek

http://www.ucy.ac.cy/%7Egeorgios/bookfiles/MATLABbook.pdf

Course Title Multivariable Differential calculus Course Code MAS 201 Course Type Compulsory Level Undergraduate Year / Semester 2nd year/3rd semester Teacher’s Name


0


To introduce students to fundamental concepts of multivariable calculus and provide them with skills allowing them to solve mathematical problems arising in the world around them.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Prove basic topological properties for ℝn and its subsets. • Prove the continuity and differentiability of multivariable functions and their

properties. • Apply the chain rule to multivariable functions. • Compute the linear approximation and Taylor polynomial of multivariable

functions, and be able to demonstrate their analyticity. • Describe the critical points of a function and find its absolute and conditional

extrema (including the method of Lagrange multipliers). • Apply and understand the inverse function Theorem and the implicit function

Theorem. Prerequisites There are no prerequisites for

this course. Required There are no co-requisites for

this course.

Course Content Normed spaces: examples, Rn, equivalent norms, Cauchy-Schwartz inequality. Open and closed sets. Compactness (Heine-Borel, Bolzano- Weierstrass Theorems). Scalar and vector valued functions. Limits and continuity. Partial derivatives. Differentiability Criterion. Multivariable functions: gradient, differential, directional derivative, vector fields, divergence, curl, Laplacian operator. Vector functions of one variable: derivatives, arc length, change of parameter. Differentiation rules, chain rule, etc. Mean value Theorem. Derivatives of integrals with respect to a parameter. Taylor’s Theorem. Local extrema, conditional extrema (Lagrange multipliers). Inverse and implicit function Theorems.

Teaching Methodology Lectures (4 hours per week) Bibliography Ν. Δανίκα, Μ. Μαριά, Μαθήματα Διαφορικού Λογισμού πολλών μεταβλητών, Εκδόσεις

Ζήτη. J. Marsden, A. Tromba, Vector Calculus, Freeman. Τ. Χατζηαφράτη, Απειροστικός Λογισμός σε πολλές μεταβλητές, Εκδόσεις Συμμετρία. J. Hubbard, B. Hubbard, Vector Calculus Linear Algebra and Differential Forms, Matrix Editions. W. Spivak, Calculus on Manifolds, Westview Press. Rudin, Principles of Mathematical Analysis, McGraw-Hill

Assessment Final Exam, 2 Midterms. Language Greek

Course Title Multivariable Integral calculus

Course Code MAS 202


Level Undergraduate

Year / Semester 2nd year/4th semester

Teacher’s Name


0


The main objective of the course is to introduce students to fundamental theorems regarding the integrability criteria and properties of the integrals of scalar and vector valued functions over Euclidean space, ℝn.


• Prove the Riemann integrability of multivariable functions. • Understand the importance of Fubini’s Theoren in the computation of an integral via

iterated integrals, and to apply it for the computation of volumes and classical integrals over regions of ℝ𝑛𝑛.

• Understand and apply the change of variable Theorem in the computation of an integral.

• Parametrize curves and surfaces and compute integrals over them. • Integrate differential forms over appropriate subsets of ℝ𝑛𝑛, use the Theorems of

Stokes, Green and Gauss in their computation, and understand their applications.



Course Content Integrable functions and sets, properties. Fubini’s Theorem. Iterated integrals for continuous functions over a compact set (scalar functions over regions of the type Q = I1 × I2 × I3 … × In). Change of variables Theorem for linear and C1-invertible transformations. Computation of volumes, Cavalieri’s principle, examples such as the sphere, cylinder and cone. Convergence theorems (interchanging limits and integrals). Transform Theorem (without proof), applications. Parametrized surfaces, partition of unity. Surface and line integrals, computing the area of a surface. Differential forms, Stokes’ Theorem (Green, Gauss, Stokes), applications.


Bibliography Τ. Χατζηαφράτη, Απειροστικός Λογισμός σε πολλές μεταβλητές, Εκδόσεις Συμμετρία. J. Hubbard, B. Hubbard, Vector Calculus Linear Algebra and Differential Forms, Matrix Editions. J. Marsden, A. Tromba, Vector Calculus, Freeman. W. Spivak, Calculus on Manifolds, Westview Press.


Language Greek

Course Title Ordinary Differential Equations

Course Code MAS 203


Level Undergraduate


Teacher’s Name


0


The purpose of this course is the study of Ordinary Differential Equations (ODEs) and their applications. The objective is to teach the students the methods for the analytical solution of ODEs as well as proving certain important theorems which govern the existence of solutions.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Model simple physical problems (from engineering, biology, economics etc) with

Ordinary Differential Equations (ODEs). • Obtain analytical solutions for 1st order ODEs (linear and non-linear) as well as for

2nd order ODEs with constant coefficients (homogeneous or not). • Construct the solution to certain ODEs (with or without singular points) using

power series. • Convert higher order ODEs to a 1st order system of ODEs and obtain its (analytical)

solution. • Handle relevant theoretical issues (such as existence and uniqueness of the

solution).



Course Content Separable ODEs. First order ODEs and integrating factors. Picard-Lindelöf theorem. Second order ODEs with constant coefficients. The method of undetermined coefficients and the method of variation of parameters. Systems of first order ODEs.


Bibliography Γ. Σ. Σμυρλής, Συνήθεις Διαφορικές Εξισώσεις, Πανεπιστήμιο Κύπρου, 2008. Δ. Σουρλάς, Συνήθεις Διαφορικές Εξισώσεις, Πανεπιστήμιο Πατρών, 2007. W. E. Boyce and R. C. DiPrima, Elementary Differential Equations, Wiley, 2009. K. R. Nagle, E. B. Saff, D. A. Snider, Fundamentals of Differential Equations, Pearson, 2003.


Language Greek

http://www2.ucy.ac.cy/%7Exenophon/courses/mas203/pdf/ode.pdf

http://www2.ucy.ac.cy/%7Exenophon/courses/mas203/pdf/ode2.pdf

Course Title Number Theory

Course Code MAS 222

Course Type Elective

Level Undergraduate

Year / Semester 4th year/7th semester

Teacher’s Name


0


To learn the basic notions of elementary Number Theory.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Understand the basic notions of Number Theory • Recognize connections between Number Theory and other areas of mathematics. • Understand the basic Theorems and the techniques of their proofs.



Course Content Divisibility, Euclidean algorithm, linear Diophantine equations. Prime numbers and the fundamental theorem of arithmetic. Congruences and the Chinese remainder theorem. Fermat’s and Wilson’s theorems. Arithmetic functions and perfect numbers. Euler’s theorem. Quadratic Reciprocity. Pell’s equation and continued fractions.


Bibliography 1. David Burton, Elementary Number Theory, McGraw Hill, 5th edition 2. Joseph Silverman, A friendly introduction to Number Theory, Pearson.


Language Greek

Course Title Probability I

Course Code MAS 261


Level Undergraduate


Teacher’s Name



To learn the basic notions of probability theory.


• Understand the basic notions of probability theory.

• Identify and apply the right tools and methodology for solving probabilistic problems.

Prerequisites A course in Mathematical Analysis


Course Content Counting methods, combinatorics, probability measure space though σ-algebras, independence of events, random variables, cumulative distribution function, discrete and continuous random variables, mean value, multivariable distributions, multivariable normal distribution, sums of random variables, distributions of functions of random variables, covariance function, independence of random variables through the cumulative distribution function, moment generating function, characteristic function, introduction to the law of large numbers, introduction to the central limit theorem.

Teaching Methodology Lectures (4 hours/week,) Tutorials (1 hour per week)

Bibliography Grimmett and Welsh. Probability an Introduction, Oxford Science Publications, 2011

Assessment Final exam, Midterm exams

Language Greek

Course Title Statistics I

Course Code MAS 262


Level Undergraduate


Teacher’s Name


0

Course Purpose and Objectives To give students the necessary background in mathematical statistics


• Understand the basic notions of statistical inference. • Identify and apply the right tools and methodologies for solving statistical inference

problems.

Prerequisites A course in Probability Required There are no co-requisites for this course.

Course Content Random samples, statistical experiments, statistics, estimation methods (e.g., method of moments, method of maximum likelihood), properties of estimators (e.g., unbiasedness, sufficiency, completeness), exponential families, Rao-Blackwell theorem, Lehmann-Scheffe theorem, Cramer-Rao variance lower bound, confidence intervals, minimum length confidence intervals, hypotheses testing, properties of tests. Statistics, sufficiency, completeness, exponential families, unbiasedness, uniformly minimum variance unbiased estimators, Cramer-Rao lower bound, moment estimators, maximum likelihood estimators, confidence intervals, hypothesis testing.

Teaching Methodology Lectures (4 hours/week,) Tutorials (1 hour per week)

Bibliography George Casella and Roger Berger. Statistical Inference, Duxbury Press, 1990

Assessment Final exam, Midterm exams

Language Greek

Course Title Numerical Analysis I

Course Code MAS 271


Level Undergraduate


Teacher’s Name


0


Introduction to Numerical Analysis


• Choose an appropriate numerical method for the approximation to the solution of non-linear equations.

• Be familiar with direct and iterative methods for solving linear systems • Construct the interpolating polynomial for a given function. • Approximate definite integrals. • Analyze the error in the aforementioned approximations as well as implement the

methods on a computer.



Course Content Sources and propagation of error. Numerical solution of non-linear equations. Numerical solution of linear systems of equations. Polynomial interpolation. Numerical quadrature.


Bibliography Γ. Δ. Ακρίβης και Β. Α. Δουγαλής, Εισαγωγή στην Αριθμητική Ανάλυση, Πανεπιστημιακές Εκδόσεις Κρήτης, 1998. K. E. Atkinson, An Introduction to Numerical Analysis, 2nd Ed., John Wiley & Sons, New York, 1988. J. F. Epperson, An Introduction to Numerical Methods and Analysis, John Wiley & Sons, New York, 2002.

Assessment 2 Midterm exams, final exam

Language Greek

Course Title Real Analysis

Course Code MAS 301


Level Undergraduate

Year / Semester 3rd year/5th semester

Teacher’s Name


1


Understand the basic concepts of Real Analysis and Topology of Metric spaces

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Understand the basic concepts of Real Analysis and the Topology of Metric spaces. • Know and apply the main Theorems of Real Analysis in relation to the compactness

of Metric spaces. • Examine the uniform convergence of sequences of functions. • Derive and integrate series of functions.



Course Content Metric spaces, Normed spaces. Examples. Open and closed sets, interior and closure of a set. Accumulation points and the derived set. The Bolzano-Weierstrass Theorem. Convergence of sequences in metric spaces. Cauchy sequences. Complete metric spaces. The fixed point theorem. Compact sets in metric spaces. The Heine-Borel Theorem. Compact metric spaces. Continuous functions. Continuous and uniformly continuous functions. Continuity and compactness. Sequences and series of functions. Uniform convergence. Uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation. The metric of uniform convergence. Sufficient conditions for uniform convergence of a series of functions.


Bibliography 1. Walter Rudin, Αρχές Μαθηματικής Αναλύσεως. Μετάφραση Δημοσθένης Σταλίδης, Εκδόσεις Leader Books, Αθήνα 2000. 2. Gerald B. Folland, Real Analysis, Modern Techniques and Their Applications, Second Edition, John Wiley and Sons, Inc., 1999. 3. Manfred Stoll. Introduction to Real Analysis, Second Edition, Addison Wesley, 2001. 4. Tom Apostol. Mathematical Analysis. Second Edition, Addison Wesley publishing company 1985.

Assessment Final Exam and Midterm Exam

Language Greek

Course Title Complex variables I

Course Code MAS 302


Level Undergraduate

Year / Semester 3rd year/ 6th semester

Teacher’s Name


0


To learn some fundamental facts about complex numbers and holomorphic (analytic) functions. To learn how to compute complex integrals using residue theory.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the integral approach of Cauchy. • Comprehend the connection with the power series approach of Weierstrass • Comprehend the geometric approach of Riemann.



Course Content Complex numbers, analytic functions, Cauchy-Riemann equations. Harmonic functions. Exponential, trigonometric and logarithmic functions. Integration, Cauchy’s theorem, Cauchy’s integral formulas and inequalities. Liouville theorem and the fundamental theorem of Algebra. Maximum modulus principle. Taylor and Laurent series, residues. The argument principle. Conformal mappings and Mobius transformations.


Bibliography 1. Knopp, K. Theory of Functions. New York: Dover, 1945. v. 2. 2. Boas, Ralph Philip. An invitation to complex analysis. Random House. 3. Fisher, Stephen D. Complex variables (Dover). 4. Μιγαδικές συναρτήσεις και εφαρμογές. Brown and Churchill, Πανεπιστημιακές

Εκδόσεις Κρήτης.

Assessment Two midterm exams plus a final exam.

Language Greek

Course Title Partial Differential Equations

Course Code MAS 303


Level Undergraduate


Teacher’s Name


0


The course objective is the introduction to the study of Partial Differential Equations (PDEs) and the construction of analytical solutions to some of them, using the method of separation of variables.

Learning Outcomes Upon successful completion of this course, students are expected to be able of:

• Classification of PDEs (eg linear or not, with respect to order, hyperbolic, parabolic, elliptic etc.)

• Proof of existence and uniqueness of the solution at a given boundary value problem and / or initial value problem.

• Proof of properties of solutions without finding analytical representations using techniques (for instance) Domain of Dependence, Maximum Principle, Hopf’s Lemma etc.

• Construction of solutions using Fourier series for given boundary value problems and / or initial value problems for the wave equation, the heat equation and the equation of Laplace.

• Construction of solutions using the method of characteristics.



Course Content 1st order PDEs, Non-linear 1st order PDEs, Linear PDEs of 2nd order, Elliptic, Parabolic, Hyperbolic PDEs, Separation of variables, Fourier series.


Bibliography Γ. Δ. Ακρίβης και Ν. Αλικάκος, Μερικές Διαφορικές Εξισώσεις, Σύγχρονη Εκδοτική. Αθήνα 2012. Γ. Δάσιος και Κ. Κυριάκη, Μερικές Διαφορικές Εξισώσεις, Αθήνα, 1994. Σ. Α. Τερσένοβ, Διαφορικές Εξισώσεις με Μερικές Παραγώγους, Εκδόσεις Συμμετρία, Αθήνα, 1992. Σ. Τραχανάς, Μερικές Διαφορικές Εξισώσεις: Σειρές Fourier και Προβλήματα Συνοριακών Τιμών, Πανεπιστημιακές Εκδόσεις Κρήτης, Ηράκλειο, 2004.

Assessment Midterm Exam, Final Exam

Language Greek

Course Title Functional Analysis

Course Code MAS 304


Level Undergraduate


Teacher’s Name


0


To become familiar with and be in a position to apply some of the basic theorems of Functional Analysis.

Learning Outcomes Upon successful completion of this course, students are expected to: • Be familiar with the basic theory of Banach spaces. • Know the Hahn-Banach Theorem, the Open Mapping Theorem, the Closed Graph

Theorem and the Principal of Uniform Boundedness and they will be able to use them in various applications.

• Know the basic theory of Hilbert spaces.



Course Content Metric and normed linear spaces, examples, series, Schauder bases, bounded linear operators, linear functionals, dual spaces. Inner product spaces, orthogonality, orthonormal sets, Bessel’s inequality, Hilbert spaces, projections, orthogonal complements. Riesz Representation Theorem, orthonormal bases. Zorn’s Lemma, Hahn-Banach Theorem with applications, the Principal of Uniform Boundedness with applications, the Open Mapping Theorem with applications, the Closed Graph Theorem with applications.


Bibliography Functional Analysis Notes, Apostolos Giannopoulos (http://www.math.uoc.gr/~nikosf/Synarthsiakh-10/Functional.pdf) Introductory Functional Analysis with Applications, Kreyszig, Wiley.

Assessment Midterm exam, final exam

Language Greek

http://www.math.uoc.gr/%7Enikosf/Synarthsiakh-10/Functional.pdf

Course Title Introduction to Algebra

Course Code MAS 321

Course Type Compulsory for students in Pure Mathematics, Elective for the rest

Level Undergraduate


Teacher’s Name


0


To learn the basic notions of group and ring theory.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Understand basic notions of Algebra. • Connect notions of algebra with other areas of mathematics. • Understand the basic Theorems and the techniques of their proofs and be able to

apply them to solving relevant problems. • Understand various applications, e.g., in Number Theory.



Course Content Groups, permutations and symmetric groups, cyclic groups. Subgroups and the Theorem of Lagrange. Homomorphisms and Quotient groups. Rings, integral domains and fields. Homomorphisms, ideals and quotient rings. Polynomial rings, divisibility in polynomial rings, prime and maximal ideals. Finite fields and field extensions.


Bibliography 1. I. N. Hersein, Abstract Algebra, Third Edition, Prentice Hall 2. Joseph J. Rotman, A First course in abstract algebra, Third edition, Prentice Hall 3. Serge Lang,Undergraduate Algebra, Springer-Verlag.

Assessment Final Exam and midterm exam

Language Greek

Course Title Classical Differential Geometry

Course Code MAS 331


Level Undergraduate


Teacher’s Name


0


To learn the basic theory of Classical Differential Geometry.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Compute the curvature of curves in the plane and in space and understand the

fundamental theorems. • Understand the notion of Gauss curvature of surfaces and compute it. • Write down the equations of geodesics of surfaces. • Understand and apply the theorem of Gauss-Bonnet.



Course Content The Euclidean space Rn: inner product, Cauchy-Schwarz inequality, isometries. Curves in Rn: parametized cuves, length, periodic, closed cuvres. Curves in R2: curvature, Frenet equalities, winding number, isoperimetric inequality, Hopf Theorem. Curves in R3: curvature, torsion, Frenet equalities, Fundamental theorem. Surfaces in R3: regular surfaces, local parametrization, examples. Differentiable maps between surfaces, tangent space, total differential. First fundamental form, orientation, Gauss map, second fundamental form, principal curvatures, curvature lines, normal curvature, Gauss curvature, mean curvature. Integration on surfaces. Ruled, minimal surfaces, surfaces of revolution. Isometric (locally isometric) surfaces, Christoffel symbols, Theorema Egregium (Gauss). Parallel vector fields, geodesics, geodesic curvature. Gauss-Bonnet Theorem.


Bibliography Lecture notes or Differential Geometry of Curves and Surfaces, Manfredo Do Carmo.


Language Greek

Course Title Stochastic Processes

Course Code MAS 350

Course Type Compulsory for the students in Probability/Statistics, Elective for the rest.

Level Undergraduate


Teacher’s Name


0


Introduction to basic concepts of stochastic processes and investigating special families of stochastic processes


• Comprehend basic concepts related to stochastic processes and of specific processes like Markov chains, Poisson process and Brownian motion.

Prerequisites Probability I and II Required

Course Content Stochastic process, stationary processes, stopping times. Markov chains, Poisson processes, Brownian motion.

Teaching Methodology Lectures and Exercises

Bibliography Theory of Stochastic Processes, Dimitris Kostantinidis, Athens 2009

Assessment Midterms and Final Exam

Language Greek

Course Title Probability II

Course Code MAS 361


Level Undergraduate


Teacher’s Name


1


To give students the necessary measure based background in probability

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the mathematical concepts of probability theory. • Use the necessary mathematical tools needed for establishing certain theoretical

results in probability theory.

Prerequisites A course in probability, a course in mathematical analysis


Course Content Review of basic elements from MAS 261. Stochastic independence through σ-algebras, Borel-Cantelli lemmas, Kolmogorov 0-1 laws, mean value as Lebesgue integral, basic inequalities, convergence of sequences of random variables, convergence of series of random variables, laws of large numbers, central limit theorems, conditional probability, conditional mean value, introduction to martingales, central limit theorem for martingales.


Bibliography Alan Gut. Probability: A Graduate Course, Springer, 2005

Assessment Final Exam, 2 Midterm exams.

Language Greek

Course Title Statistics II

Course Code MAS 362


Level Undergraduate


Teacher’s Name



The students should be able to comprehend the mathematical concepts of statistical inference.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Use the necessary mathematical tools needed for establishing certain theoretical

results in statistical inference.

Prerequisites A course in probability, a course in statistics, a course in mathematical analysis


Course Content Asymptotic properties of estimators, asymptotic efficiency, asymptotic normality, introduction to statistical decision theory (minimax estimators, Bayes estimators), asymptotic properties of tests, optimal tests, goodness-of-fit tests, tests of independence. U-statistics.


Bibliography Robert Keener. Theoretical Statistics, Springer, 2010

Assessment Final Exam, Midterm Exam.

Language Greek

Course Title Numerical Analysis II

Course Code MAS 371


Level Undergraduate

Year / Semester 3rd or 4th year of study

Teacher’s Name


0


Advanced course in numerical analysis (eigenproblems, iterative solution of linear systems, interpolation, orthogonal polynomials, quadrature).

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Know the basic theory and solve problems related to the theory of eigenvalues and

eigenvectors. • Know the basic theory and solve problems related to vector and matrix norms. • Know the basic theory of iterative methods for solving linear systems. In particular

Jacobi, Gauss-Seidel, JOR and SOR. • To know the latest numerical methods for the calculation of eigenvalues and

eigenvectors. • Use Lagrange and Hermite interpolation. • Know the basic theory of orthogonal polynomials on the real line. • To construct and apply Gaussian integration formulae.



Course Content Brief revision of the theory of eigenvalues and eigenvectors. Positive definite matrices. Vector and matrix norms. Iterative methods for the solution of linear systems. Gershgorin bounds for eigenvalues. Numerical methods for eigenvalues and eigenvectors. Lagrange interpolation. Hermite interpolation. Divided differences at repeated points. The Newton form of the Hermite interpolation polynomial. Orthogonal polynomials. Gaussian quadrature.


Bibliography An Introduction to Numerical Analysis, K. Atkinson. Εισαγωγή στην Αριθμητική Ανάλυση, Γ. Ακρίβης - Β. Δουγαλής. Introduction to Numerical Analysis, J. Stoer and R. Bulirsch. Fundamentals of Matrix Analysis, E. Saff - A. Snider.

Assessment Two midterm exams, one computational project, final exam.

Language Greek

Course Title Measure Theory and Integration

Course Code MAS 401


Level Undergraduate

Year / Semester 4th year students

Teacher’s Name


0


Understand basic concepts of Measure Theory and Integration

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Understand basic concepts of Measure Theory and Integration. • Prove the main results of Measure Theory and Integration. • Conduct rigorous mathematical arguments and reasoning when solving problems in

Measure Theory and Integration.



Course Content General revision: Sets, orderings, cardinality, metric spaces. Measures: Algebras and σ-algebras, additive and σ- additive measures, outer measures, Borel measures on the real line. Integration: measurable functions, integration of positive functions, integration of complex valued functions, Convergence Theorems, modes of convergence, product measures, the n – dimensional Lebesgue integral, integration in polar coordinates, signed measures, the Radon – Nikodym theorem, complex measures, differentiation on Euclidean space, functions of bounded variation. Lp Spaces: The basic theory, the dual of Lp, the useful inequalities, the distribution function, weak – Lp spaces, interpolation.


Bibliography Γ. Κουμούλης, Σ. Νεγρεπόντης, Θεωρία Μέτρου, Εκδόσεις Συμμετρία, Αθήνα 1988. Gerald B. Folland, Real Analysis, Modern Techniques and Their Applications, Second Edition, John Wiley and Sons, Inc., 1999. Robert G. Bartle, The Elements of Integration and Lebesgue Measure, John Wiley and Sons, Inc., 1995. H. L. Royden, Real Analysis, third edition, Macmillan Publishing Company, 1988. Elias M. Stein and Rami Shakarchi: Real Analysis, Measure Theory, Integration and Hilbert Spaces. Princeton Lectures in Analysis III. Princeton University Press 2005.

Assessment Final Exam, Midterm Exam.

Language Greek

Course Title Complex Analysis II

Course Code MAS 402


Level Undergraduate

Year / Semester 4th year

Teacher’s Name The course has not been offered at the time


0


To learn more advanced topics from Complex Analysis and its connection to other scientific fields (namely Physics, through Riemann surfaces)

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Understand complex analysis in depth. • Make connections with geometric theory. • Understand the basic Theorems and the techniques of their proofs.

Prerequisites MAS 302 Required There are no co-requisites for this course.

Course Content Compactness and convergence in the space of analytic functions. The space of meromorphic functions. Riemann mapping theorem. Weierstrass Theorem on entire functions, analytic continuation. Elliptic functions. Riemann surfaces.


Bibliography 1. Complex Functions, G. Jones and D. Silverman, Cambridge 2. Markushevich, Theory of functions, Chelsea.

Assessment Homework (5 sets of problems) and a final exam

Language Greek

Course Title Ordinary Differential Equations II

Course Code MAS 403


Level Undergraduate

Year / Semester 3rd or 4th year



0


Presentation of the theory of ODEs and their applications

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Understand basic concepts of the theory of ODEs • Prove main results of the theory • Conduct rigorous mathematical arguments and reasoning when solving problems

and exercises.



Course Content Boundary Values Problems. Sturm-Liouville type problems. Asymptotic behaviour of nonlinear systems of ODEs: Stability. Perturbation Theory of systems of ODEs which possess periodic solutions. Perturbations of two-dimensional autonomous systems. Poincaré-Bendixson theory.


Bibliography F. Hildebrand, Methods of Applied Mathematics, Dover Publ New Edition 1992. E. Coddington & N. Levinson, Theory of ODEs, McGraw-Hill 1955.

Assessment Final Exam, Midterm Exam

Language Greek

Course Title An Introduction to Fourier Analysis

Course Code MAS 418


Level Undergraduate

Year / Semester 3th or 4th year



0


Presentation of the basic theory of Fourier Series.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Understand basic concepts of Fourier Analysis. • Prove main Theorems and formulas of Fourier Analysis. • Conduct rigorous mathematical arguments and reasoning when solving problems in

Fourier Analysis. • Present applications of Fourier Analysis in Physics and Signal Processing.



Course Content Periodic functions, trigonometric polynomials, trigonometric series. Fourier series. Convergence of Fourier series. Bessel’s inequality. Completness, Parseval’s Theorem. The Riemann-Lebesgue Lemma. Dirichlet’s Theorem. Gibbs phenomenon. Differentiation and Integration of Fourier series. Cesaro and Abel summability of Fourier series. Fejer’s Theorem. Poisson’s Theorem. The Fourier transform and its properties. The inversion theorem and Plancherel’s identity. The convolution and its properties. Applications to partial differential equations.


Bibliography Georgi P. Tolstov, FOURIER SERIES, (translated from the Russian by Richard A. Silverman), Dover publications, 1976. Allan Pinkus and Samy Zafrany, FOURIER SERIES AND INTEGRAL TRANSFORMS, Cambridge University Press, 1997. Gerald B. Folland, FOURIER ANALYSIS AND ITS APPLICATIONS, Brooks/Cole Publishing company, 1992. A. Zygmund, TRIGONOMETRIC SERIES, 3rd edition, Cambridge University Press, Cambridge, 2002.

Assessment Final Exam, 2 Midterm exams

Language Greek

Course Title Special Topics In Mathematical Analysis

Course Code MAS 419


Level Undergraduate




0


To introduce students to subjects related to research interests/topics of the academic staff of the department from the area of Mathematical Analysis.

Learning Outcomes This is a course intended to introduce students to research in mathematics, through reading some classical papers

Prerequisites MAS 301, MAS 302 Required

Course Content Depends on the special interest of the staff member teaching it.

Teaching Methodology Lectures (4 hours per week), Arranged meetings with the staff member teaching it.

Bibliography Depends on the special interest of the staff member teaching it.

Assessment Homework (problem sets), presentation.

Language Greek

Course Title Approximation Theory

Course Code ΜΑS420


Level Undergraduate




0


The main course purpose is the introduction of students to approximation theory in normed linear spaces and inner product spaces. The course objective is the understanding of various methods of the theory and the mathematical analysis for best approximation.


• Recognize metric spaces and linear normed spaces. • Prove estimates using norms • Apply fixed point techniques to prove theorems. • Use iterative methods to solve linear systems • Find the best approximation in metric, normed or inner-product spaces.



Course Content Normed linear spaces and inner product spaces. Bounded linear operators. Fixed Point methods. Iterative methods for linear systems. Best approximation in normed linear spaces and inner product spaces. Orthogonal polynomials.

Teaching Methodology Lectures and exercises

Bibliography Introduction to Approximation theory, by E.W. Cheney, American Mathematical Society, Providence, RI 1982.

Approximation Theory and Methods, by M.J.D. Powell. Cambridge University Press, Cambridge, 1981.

Assessment One final and one midterm exam

Language Greek

Course Title Introduction to Coding Theory

Course Code MAS422


Level Undergraduate




0


Introduction to basic notions of coding theory with examples of codes


• Understand the problem of decoding and principles of decoding. • Encode and decode with a linear code (error correcting and error detecting,

syndromes). • Understand the examples of cyclic codes.



Course Content Introduction to finite fields. Vector spaces over finite fields. Linear codes. Encoding and decoding with a linear code. Syndrome decoding. Cyclic codes. Hamming codes.


Bibliography O. Pretzel, Error-Correcting Codes and Finite Fields.

Δ. Βάρσος, Στοιχεία Αλγεβρικής Θεωρίας Κωδίκων

Assessment Final exam, 2 Midterm exams

Language Greek

Course Title Theory of Rings and Modules

Course Code MAS 424


Level Undergraduate




0


Introduction to basic concepts and results from the theory of rings and modules as well as applications

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the importance of a ring as a fundamental object in algebra. • Comprehend the concept of a module as a generalisation of a vector space and an

Abelian group. • Understand the proofs and some immediate applications of the basic theorems. • Know fundamental constructions and be familiar with basic properties of Noetherian

rings.



Course Content Rings and ideals. Homomorphism Theorems. Unique factorisation domains and principal ideal domains. Factor rings. Prime and maximal ideals. R-modules and homomorphisms. Finitely generated R-modules. Noetherian rings.

Teaching Methodology Lectures and exercises (4 hours per week)

Bibliography 1. M.E. Keating. A first course in module theory. Imperial College Press, London, 1998. 2. J.J. Rotman. A First Course in Abstract Algebra, With Applications, 3rd edition,

Pearson Prentice Hall, 2006

Assessment Final Exam and Midterm exam

Language Greek

Course Title Group Theory

Course Code MAS 425


Level Undergraduate




0


To learn the basic theory of groups.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Understand basic notions in Group Theory. • Make connections of certain group theoretic notions with other areas of Mathematics. • Acquire a sound understanding of the basic theorems and techniques involved in their

proofs and also apply those in the solution of relating problems. • Study several applications of the theorems, for example in finite group classification

problems.



Course Content Normal subgroups, homomorphism theorems. Direct and semidirect products. Group actions. Normalizers and centralizers. Sylow theorems and p-groups. Simple groups. Finitely generated Abelian groups. Composition series and Jordan – Hölder theorem. Soluble groups.


Bibliography W. Ledermann, A. Weir, Introduction to Group Theory. J.J. Rotman, An Introduction to the Theory of Groups.

Assessment Two midterms and one final exam.

Language Greek

Course Title Galois Theory

Course Code MAS 426


Level Undergraduate


Instructor’s Name


0


The understanding of basic notions of ring and field theory and how this theory can be used in order to solve classical ancient problems like the construction of regular polygons and squaring the circle.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Acquire a sound understanding of basic notions of Field Theory with an application of

those for the solution of some of the classical ancient problems such as squaring the circle, trisecting an angle and constructing regular polygons which were eventually solved using methods of modern Field Theory.

• Acquire the ability of solving problems in this area (including computations such as the Galois group of a polynomial or the Galois correspondence in a number of simple examples).



Course Content Rings, ideals, polynomial rings. Fields, field extensions, algebraically closed fields, finite fields. Normal extensions and Galois extensions. The fundamental theorem of Galois theory. Solutions of equations by radicals, ruler and compass constructions.


Bibliography J.J. Rotman, Galois Theory. I.N. Herstein, Topics in Algebra. Σ.Α. Ανδρεαδάκη, Θεωρία Galois.

Assessment Final Exam

Language Greek

Course Title Group Representation Theory

Course Code MAS 427


Level Undergraduate




0


Introduction to basic concepts and results from the theory of group representations as well as applications

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Know the statements and understand the proofs of the basic theorems. • Be able to compute character tables of groups. • Understand basic concepts of the representation theory of finite groups over the

complex numbers and their application in solving problems. • Learn about the connection of this theory with other areas of mathematics



Course Content Representations. FG - modules, FG - submodules, FG - homomorphisms. Maschke’s Theorem and Schur’s Lemma. Irreducible modules. The group algebra, the centre of the group algebra. Characters, relation between characters and representations.

Teaching Methodology Lectures and Exercises (4 hours per week)

Bibliography G. D. James and M. Liebeck, Representations and Characters of Finite Groups. 2nd edition, Cambridge University Press, 2001.

Assessment Final Exam

Language Greek

Course Title Topics In Algebra

Course Code MAS 429


Level Undergraduate




0


To introduce students to subjects related to research interests/topics of the academic staff of the department from the area of Algebra.

Learning Outcomes This is a course intended to introduce students to research in mathematics, through reading of some classical papers



Course Content Depends on the special interest of the staff member teaching it


Bibliography Depends on the special interest of the staff member teaching it.

Assessment Homeworks, final exam.

Language Greek

Course Title Introduction to Differentiable Manifolds

Course Code MAS 431


Level Undergraduate




0


To learn the basic theory of differentiable manifolds

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Understand the notion of differentiable manifolds and transfer methods from

multivariate calculus to manifolds • Perform computations with differential forms • Understand Stoke΄s Theorem.



Course Content Differentiable manifolds. Tangent space. Partition of unity. Sards Theorem. Vector fileds, flows. Frobenius theorem. Differential forms. Theorem of Stokes. Theorem of de Rham.


Bibliography • Lecture notes • W. M. Boothby, An Introduction to differentiable Manifolds and

Riemannian Geometry. • John Lee, Introduction to Smooth Manifolds.

Assessment Two midterm exams plus a final exam.

Language Greek

Course Title Introduction to Riemannian Geometry

Course Code MAS 432


Level Undergraduate

Year / Semester 4th Year

Teacher’s Name


0


The main objective of the course is to introduce students to the fundamental definitions and theorems of Riemannian Geometry.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Provide atlases for classic manifolds. • Find the geodesic equation on a Riemannian manifold and compute the curvature of

the manifold. • Prove properties of Riemannian manifolds using the exponential map and Jacobi

fields.



Course Content Introduction to manifolds, tangent spaces and vector fields. Riemannian manifolds. Connections, geodesics, exponential map, normal coordinates, Gauss’ Lemma. Hopf-Rinow Theorem. Curvature. Jacobi fields. Theorems of Bonnet-Myers, Synge-Weinstein and Hadamard-Cartan.


Bibliography M. Do Carmo, Riemannian Geometry, Birkhauser, 1992. J. M. Lee, Riemannian Geometry, Springer 1997 W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, 1986. M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1, Vol. 2, Publish or Perish, 1999.

Assessment Midterm and Final Exam

Language Greek

Course Title Topology

Course Code MAS 433


Level Undergraduate


Teacher’s Name


0


A solid understanding of basic notions of point set topology and algebraic topology and their connection to other branches of mathematics like group theory and geometry.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Acquire a solid understanding of basic notions of point set topology and algebraic

topology and their connection to other branches of mathematics like group theory and geometry.

• Acquire the ability to solve problems of the field.



Course Content Topological spaces, continuous functions, connected and compact sets, product spaces, the Tychonoff theorem, separation axioms, metric spaces. Homotopy, the fundamental group, the Seifert Van Kampen Theorem, Covering spaces.


Bibliography J. Munkres, Topology. I. M. Singer, J. A. Thorpe, Lecture Notes in Elementary Topology and Geometry. W. Massey, Basic Course in Algebraic Topology. A. Hatcher, Algebraic Topology.

Assessment Final Exam.

Language Greek

Course Title Introduction to Algebraic Topology

Course Code MAS 434


Level Undergraduate




0


A solid understanding of basic notions of algebraic topology like homology and cohomology and their connection to other branches of mathematics like group theory, algebra and geometry.


• Acquire a solid understanding of basic notions of algebraic topology like homology and cohomology and their connection to other branches of mathematics like group theory, algebra and geometry.

• Acquire the ability to solve problems of the field.

Prerequisites MAS 321, MAS 433 Required There are no co-requisites for this course.

Course Content Homology, Cohomology. CW complexes. Homology and Cohomology of product spaces, the Eilenberg Zielber theorem and the Kunneth formula. The cohomology ring. Cup product. Poincare duality. Applications: Homology and cohomology of compact surfaces, the Jordan Brouwer separation theorem, invariance of domain.


Bibliography W. Massey, Basic Course in Algebraic Topology. A. Hatcher, Algebraic Topology.

Assessment Midterm and Final Exam

Language Greek

Course Title Introduction to Algebraic Geometry

Course Code MAS 439


Level Undergraduate




0


To introduce students to subjects related to research interests/topics of the academic staff of the department from the area of Geometry, such as Differential Geometry, Algebraic Geometry and Algebraic Topology.

Learning Outcomes Introduce students to mathematical research through the study of some classical papers and results, and to improve their ability to solve problems in the field.

Prerequisites There are no pre-requisites for this course.


Course Content Depends on the special interests of the staff member teaching it.


Bibliography Depends on the special interests of the staff member teaching it.

Assessment Homework (problem sets), presentation, final exam

Language Greek

Course Title Linear Models I

Course Code MAS 451


Level Undergraduate


Teacher’s Name


1


Introduction to regression models and analysis of variance for undergraduate students.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Know introductory data modeling methods and their immediate application to

software. • Know how to present the results, inferences, hypotheses testing, and evaluate the

suitability of the model.

Prerequisites Basic courses in Probability and Statistics, a course in Calculus and a course in Linear Algebra.


Course Content Simple linear regression model: estimation, confidence intervals, hypothesis testing. Multiple linear regression model: estimation, confidence intervals, hypothesis testing. Goodness of fit, residual analysis and model selection. One and two-way ANOVA.


Bibliography Applied Linear Statistical Models, by Netter et all., McGraw Hills. Applied Regression Analysis, by Draper & Smith, Wiley.

Assessment Final exam, midterm exams, data analysis projects.

Language Greek

Course Title Linear Models II

Course Code MAS 452


Level Undergraduate


Teacher’s Name


0


Extension of the theory of linear models

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Know further data modeling methods and their immediate application to software. • Know how to present the results, inferences, hypotheses testing, and evaluate the

suitability of the model.

Prerequisites Basic courses in Probability and Statistics, , a course in Calculus and a course in Linear Algebra, Linear Models I.


Course Content Analysis of variance with one or more fixed-effects, Analysis of variance with one or more random factors, analysis of covariance. Generalized linear models: estimation in some examples, logistic regression, asymptotic properties of estimators.


Bibliography Applied Linear Statistical Models, by Netter et all., McGraw Hills. Applied Regression Analysis, by Draper & Smith, Wiley. An Introduction to GLM (Second Ed), A. Dobson, Chpaman & Hall.


Language Greek

Course Title Non-Parametric Statistics

Course Code MAS 454


Level Undergraduate




0


The students should be able to comprehend the mathematical concepts of non-parametric statistical inference.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Use and apply the necessary mathematical and computational tools needed for

establishing certain theoretical results and be able to apply them in various datasets.

Prerequisites A course in probability, a course in statistics, a course in functional analysis.


Course Content Distribution function estimation, probability density function estimation, regression function estimation, applications in R.


Bibliography M.P. Wand and M.C. Jones: Kernel Smoothing, Chapman and Hall, 1995

Assessment Final exam, midterm exam.

Language Greek

Course Title Sampling Theory

Course Code MAS 455


Level Undergraduate




0


To give students the necessary background in survey methodology

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Justify mathematically all the basic steps of a sampling design. • Design all the stages of a sample survey. • Recognize all possible nonsampling errors and design the survey in such a way so

that these errors are minimized.

Prerequisites A basic course in probability and a basic course in statistics


Course Content Sampling scheme design. Simple random sampling, stratified, systematic, cluster sampling, multistage sampling. Mean and variance estimation, ratio estimators, linear regression estimators, optimal choice of sample size, bias in survey methodology.

Teaching Methodology Lectures ( 3 hours/week) Lab (1 hour/week)

Bibliography P. S. Levy and S. Lemeshow: Sampling of Populations, Wiley, 3rd ed. 1999. S. K. Thompson: Sampling, Wiley, 1992.

Assessment Final exam, midterm exam, homework assignments in R.

Language Greek

Course Title Time Series

Course Code MAS 456


Level Undergraduate




0


Introduction to time series analysis

Learning Outcomes Statistical Inference for stochastic processes.

Prerequisites Statistic I and II, Probability I and II, Stochastic Processes.


Course Content Stationary processes, autocovariance function, spectral density, linear processes, ARMA processes, non-linear processes, ARCH and GARCH processes. Estimation of the mean and of the autocovariance function. Moment estimators, least squares estimators and maximum likelihood estimators of parameters. Asymptotic properties.


Bibliography P. Brockwell and R. Davis (1991). Time Series: Theory and Methods. Springer.

Assessment Final Exam, Midterm and Exam

Language Greek

Course Title Statistical Data Analysis

Course Code MAS 458


Level Undergraduate



ECTS 7 Lectures / week 2 (Lectures + Computer Lab)

Laboratories / week

0


Introduction to statistical computation and simulation

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Use software to perform simulations of random variables and apply simulation

techniques. • Carry out computations for summarizing data and studying their properties.

Prerequisites Basic courses in Probability and Statistics.


Course Content Introduction to R, diagnostic statistics, simulation methods, Markov chain Monte-Carlo, simulation, optimization, resampling.


Bibliography 1. Simulation, Sheldon M. Ross (Academic Press). 2. R Lecture Notes, K. Fokianos. 3. An Introduction to the Bootstrap, B. Efron and R. Tibshirani., Chapman & Hall. 4. Computational Statistics, J. Gentle, Springer.


Language Greek

Course Title Multivariate Analysis

Course Code MAS 459


Level Undergraduate




0


Introduction to the basic concepts and methodologies of multivariate analysis.

Learning Outcomes Multivariate Distributions, Statistical inference in many dimensions.

Prerequisites Statistic I and II, Probability I and II


Course Content Multivariable distributions. Mean vector and covariance matrix estimation. Wishart distribution. Principal components, canonical correlation, cluster and discriminant analysis. Testing hypothesis in many dimensions.


Bibliography T. W. Anderson (1984). An Introduction to Multivariate Statistical Analysis. Wiley R. A. Johnson and D. W. Wichern (1998). Applied Multivariate Statistical Analysis. Prentice Hall.

Assessment Midterm and final exam

Language Greek

Course Title Survival Analysis

Course Code MAS 466


Level Undergraduate




0


To give students the necessary background in survival analysis

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Comprehend the fundamental concepts of survival analysis

• Apply the theory on data to produce point and interval estimates of parameters of interest

• Perform the relevant hypotheses tests.

Prerequisites A course in probability, a course in statistics.


Course Content Censored data, Truncated data. Survival function and hazard function. Nonparametric estimation of the survival function and the hazard function. Parametric models for the hazard function. Counting processes and martingales. Semiparametric Cox model. Tests for one or more populations, tests of class-K.

Teaching Methodology Lectures (3 hours per week), tutorial (1 hour per week)

Bibliography D. Collett: Survival Analysis in Medical Research. Chapman & Hall. D. W. Hosmer, Jr., S. Lemeshow, S. May, Applied Survival Analysis, Wiley.

Assessment Final exam, midterm exams, computer projects.

Language Greek

https://www.google.com.cy/search?hl=el&tbo=p&tbm=bks&q=inauthor:%22David+W.+Hosmer,+Jr.%22

https://www.google.com.cy/search?hl=el&tbo=p&tbm=bks&q=inauthor:%22Stanley+Lemeshow%22

https://www.google.com.cy/search?hl=el&tbo=p&tbm=bks&q=inauthor:%22Susanne+May%22

Course Title Topics in Probability and Statistics I

Course Code MAS 468


Level Undregraduate

Year / Semester 3rd year

Teacher’s Name


0


To present to students various topics in probability and statistics which are not covered in the basic compulsory probability and statistics courses offered by the department

Learning Outcomes With the completion of the course, the students should be familiar with various topics in probability and statistics which are not covered in the standard probability and statistics courses offered by the department and be familiar with applications in areas such as computer science and medicine.

Prerequisites A course in probability theory and a course in mathematical statistics

Required

Course Content Topics from combinatorial theory, topics for sampling theory, topics from data analysis, classical problems from probability with applications to computer science, topics from design of experiments, Chi-square tests.


Bibliography A variety of sources

Assessment One midterm exam, homework assignments (in R), final exam

Language Greek

Course Title Topics in Probability and Statistics II

Course Code MAS 469


Level Undergraduate

Year / Semester 4th year/2nd semester

Teacher’s Name



To present to students various topics in probability and statistics that are not covered in the basic compulsory probability and statistics courses offered by the department

Learning Outcomes With the completion of the course, the students should be familiar with different topics that are not covered by the standard department courses and be familiar with applications in areas including engineering and environmental science

Prerequisites A course in probability theory and a course in mathematical statistics

Required

Course Content Stochastic processes/ fields, sample path properties, the notions of stationarity and ergodicity, covariance functions, variograms, spectra, predictions method, kriging.


Bibliography Various sources

Assessment 2 homework assignments, one midterm exam and one final exam

Language Greek

Course Title Numerical Solution of Ordinary Differential Equations

Course Code MAS 471


Level Undergraduate

Year / Semester 3rd year

Teacher’s Name


0


The purpose of this course is the introduction of standard techniques for numerically solving ordinary differential equations (ODEs).

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Solve analytically initial value problems for first and second order difference

equations. • Solve analytically and numerically initial value problems for first order ordinary

differential equations. • Solve analytically and numerically initial value problems for second order ordinary

differential equations after transforming these to first order systems. • Study the stability of one-step and multiple-step methods. • Solve analytically and numerically boundary value problems for second order

ordinary differential equations with finite difference methods.



Course Content Brief revision of initial and boundary value problems for ordinary differential equations. Stability of initial and boundary value problems for difference equations. One-step methods for initial value problems. Runge-Kutta methods. Multistep methods for initial value problems. Shooting methods and finite difference for boundary value problems.


Bibliography Σημειώσεις διαλέξεων. Α. Καραγιώργης. Αριθμητικές Μέθοδοι για Συνήθεις Διαφορικές Εξισώσεις, Γ. Ακρίβης και Β. Δουγαλής. Computer Methods for Ordinary Differential Equations and Differential Algebraic Equations, U. Ascher and L. Petzhold.

Assessment Final exam, two midterm exams, two computational projects.

Language Greek

Course Title Numerical Solution of Partial Differential Equations

Course Code MAS 472


Level Undergraduate




0


The purpose of this course is the introduction of the finite difference method for the numerical solution of partial differential equations (PDEs).

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Solve analytically and numerically initial/boundary value problems for the one-

dimensional heat equation. • Solve analytically and numerically initial/boundary value problems for the one-

dimensional wave equation. • Solve analytically and numerically boundary value problems for the two-

dimensional Poisson equation. • Study the stability of finite difference methods for the numerical solution of the

above problems.



Course Content Brief revision of initial/boundary and boundary value problems for partial differential equations. One-dimensional parabolic equations. Explicit method, implicit method and Crank-Nicolson method. Stability analysis. Two-dimensional parabolic equations. Hyperbolic equations. CFL condition. The Wendroff, Lax-Wendroff and leap frog methods. Explicit and implicit methods for the one dimensional wave equation. Elliptic equations. The five-point and nine-point methods for the Poisson equation. Non-rectangular regions. Robin boundary conditions.


Bibliography Σημειώσεις διαλέξεων. Α. Καραγιώργης και Γ. Σμυρλής. Numerical Solution of Partial Differential Equations, K. Morton and D. Mayers. Numerical Solution of Partial Differential Equations: Finite Difference Methods, G. Smith.

Assessment Final exam, two midterm exams, two computational projects.

Language Greek

Course Title Introduction to the Finite Element Method

Course Code MAS 473


Level Undergraduate


Teacher’s Name


0


The course objective is the introduction to the basic concepts and techniques of the Finite Element Method (FEM) for the numerical solution of differential equations.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • obtain the variational formulation to elliptic boundary value problems • show existence and uniqueness of the variational solution. • choose the appropriate discretization for the variational problem. • implement the method on a computer. • obtain error bounds for the discrete method.



Course Content Variational formulation of elliptic problems. Methods of Galerkin and Ritz. Basis functions and discretization. Error bounds. Applications and examples. Finite elements for parabolic equations.


Bibliography Lecture notes: Christos Xenophontos. Γ. Δ. Ακρίβης, Μέθοδοι πεπερασμένων στοιχείων, Πανεπιστημιακές Εκδόσεις, Πανεπιστήμιο Κύπρου, 2005. C. Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge University Press, 1987. D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 2001.

Assessment Midterm Exam, Final Exam, homework, project

Language Greek

http://www.mas.ucy.ac.cy/%7Exenophon/courses/mas473/pdf/nmpde.pdf

Course Title Applied Mathematical Analysis

Course Code MAS 481


Level Undergraduate

Year / Semester 4th year/ 7th semester

Teacher’s Name


0


An introduction to certain methods that can used to solve problems in Applied Mathematics.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Apply calculus of variations • Solve differential equations using Laplace transforms • Solve partial differential equations using Fourier series, Fourier transforms and

Laplace transforms • Solve special differential equations • Solve integral equations of simple forms



Course Content Calculus of variations. Laplace transforms. Fourier analysis. Special functions. Integral equations.


Bibliography F.B. Hildebrand, Methods in Applied Mathematics, Dover, 1992. M.R. Spiegel, Ανάλυση Fourier, McGraw-Hill, (Μετάφραση, Σ.Κ. Περσίδης ΕΣΠΙ, Αθήνα), 1978.

Assessment Mid-term exams and final exam

Language Greek

Course Title Classical Mechanics

Course Code MAS 482


Level Undergraduate




0


Introduction to the basic theory of Newtonian Mechanics.

Learning Outcomes Upon successful completion of this course, students are expected to be able to: • Understand Newton’s laws and their applications to the motion of a particle and a

rigid body • Understand Kepler’s laws and their application to the motion of the planets • Understand Lagrange’s equations and their applications to various problems of

Classical Mechanics



Course Content Newton’s laws. Central forces. Moving coordinate systems. Kepler’s laws. Systems of particles. Plane motion of rigid bodies. Space motion of rigid bodies. Lagrange’s equations.


Bibliography - M. Lunn, A First Course in Mechanics, Oxford Science Publications, Oxford University Press, 1996. - M.R. Spiegel, Theoretical Mechanics, Schaum's Outline Series, McGraw-Hill, 1969.

Assessment Mid-term exams and Final Exam

Language Greek

Course Title Fluid Dynamics

Course Code MAS 483


Level Undergraduate




0


Introduction to Fluid Dynamics and the equations of motion and the solution of basic flows in different coordinate systems.

Learning Outcomes Upon successful completion of this course, students are expected to: • Acquire the necessary background of vector and tensor to understand and formally

prove the equations of motion. • Be able to simplify and solve flow problems in various coordinate systems. • Understand basic notions of fluid dynamics, such as the streamfunction, the boundary

layer, non-Newtonian flows etc.



Course Content Introduction to vector and tensor calculus. Continuity and momentum equations in various coordinate systems. Laminar incompressible flows amenable to analytical solution. Steady-state and transient flows and applications. Streamfunction and Stokes flow. Boundary layer. Non-Newtonian flows.


Bibliography J. Marsden και A. Tromba, Διανυσματικός Λογισμός Πανεπιστημιακές Εκδόσεις Κρήτης, Ηράκλειο (1992). T. Papanastasiou, G. Georgiou and A. Alexandrou, Viscous Fluid Flow, CRC Press, Boca Raton, 1999.

Assessment Homeworks, two mid-exams, project with presentation.

Language Greek

Course Title Introduction to mathematical modelling

Course Code MAS484


Level Undergraduate




0


Introduction to the modeling of physical problems and familiarization with methods and software for solving the resulting mathematical problems.


• Model a physical problem to a mathematical one • Choose a proper solution method for the mathematical problem • Use available software for the numerical solution of the problem and the

visualisation of the solutions



Course Content Emphasis is given on the role of mathematical modeling as a tool for learning and understanding mathematical techniques. The applications come from various disciplines, such as discrete dynamical systems, graphs and networks, and linear programming. In this course various software packages are used extensively.


Bibliography G.B. Whitham Linear and Nonlinear Waves, Wiley, New York J.D. Logan Applied Mathematics: A contemporary approach, Wiley, New York

C.C. Lin and L.A. Segel Mathematics Applied to Deterministic Problems in the Natural Sciences, Classics in Applied Mathematics, SIAM

Assessment Mid-term exam, Final exam

Language Greek

Course Title Special Topics in Applied Mathematics

Course Code MAS487


Level Undergraduate


Teacher’s Name


0


Introduction to selected topics of Applied Mathematics.


• Approximate and normalize a physical problem • Derive approximate perturbation solutions of algebraic equations, ODEs and PDEs



Course Content 1. Dimensional analysis and normalization.

2. Perturbation methods.

3. Waves and continua


Bibliography J. David Logan, Applied Mathematics, Πανεπιστημιακές Εκδόσεις Κρήτης, Ηράκλειο, 2002.

Assessment Homeworks, two mid-exams, individual project with presentation.

Language Greek

Course Title Independent Study

Course Code MAS 499


Level Undergraduate



ECTS 7 Lectures / week Weekly Meetings (WM)

Laboratories / week

0


Introduce the student to research

Learning Outcomes The student should understand in depth a specific topic in Mathematics or Statistics

Prerequisites Successful completion of undergraduate courses

Required None

Course Content Depends on the faculty

Teaching Methodology Weekly meetings

Bibliography Depends on the topic

Assessment Delivery of written report and oral examination

Language Greek

Course Title Work Placement I, II, III (one course per month of work)

Course Code MAS501, MAS502, MAS503 (one code per month of work)

Course Type Elective (the ECTS of this course do not contribute towards the sum of 240 ECTS required for obtaining the degree

Level Undergraduate

Year / Semester Summer semester between 2nd and 3rd year of studies or summer semester between 3rd and 4th year of studies

Teacher’s Name

ECTS 1 ECTS per month of work

Lectures / week -- Laboratories / week --


- To offer the students the possibility to apply the competencies acquired during their studies at the University in real world problems, and by doing so to form a more complete view of their field (through either a specific project assigned to the students by the Hosting Organization at the start of the term of the internship, or through the day to day work in the area of activity of the Hosting Organization).

- To provide the students with industrial and/or research experience and as a result a better understanding of the job market.

- To enable students to make informed decisions relating to their career after graduating from the University.

Learning Outcomes - Application of Mathematical Skills in practical problems - Acquisition of added knowledge, Mathematical skills and communication abilities - Form better perspective of job market - Networking with potential employers

Prerequisites 120 ECTS of the Undergraduate program completed

Required --

Course Content Work placement, the responsibilities will be assigned by the hosting institution under the supervision of the academic supervisor

Teaching Methodology Work placement, at least 120 hours of work per month

Bibliography --

Assessment Timesheets and Final Evaluation Report (length 2000 words). Evaluation of work supervisor in the hosting institution (30%) and evaluation of the student’s report by the academic supervisor (70%). The student successfully completes the module if he/she achieves an evaluation of at least 5/10. The course’s final grad is Pass or Fail.

Language Greek or English

Course Title Methods for Solving Mathematical Problems

Course Code MAS 857, MAS 858


Level Undergraduate




0


To learn methods for solving mathematical problems

Learning Outcomes Throughout the course students will be presented with various techniques and consequently examples and exercises for each technique. The problems will come from different areas of mathematics, such as algebra, analysis, combination, number theory, geometry, and so on. The course will be an elective but will be taken under consideration for the selection of the students who will represent the Department in international Olympiads.



Course Content The course is intended for undergraduate students who are interested in solving mathematical problems and at the same time aims to prepare students participating in mathematical Olympiads. This course will be different from the other courses of the Department which are focusing on a specific area of mathematics. Emphasis will be on problem-solving techniques, cultivation of creative thinking. and successful presentation of solutions.

Teaching Methodology Lectures (4 hours weekly)

Bibliography Exercises from IMO, BMO, Putnam, Monthly.

Assessment Problem presentations, midterm and final exam.

Language Greek

Course Title Introduction to Programming (Fortran)

Course Code EPL031

Course Type Compulsory for students of MAS Department

Level Undergraduate

Year / Semester 1st year / 1st semester

Teacher’s Name

ECTS 7 Lectures / week 2 Χ 1.5h + 1 h Laboratories / week 1 h


Introduction of the basic principles of programming with emphasis on structured programming, abstraction, and the design, implementation, checking and debugging of modular programs. Application of these principles using the FORTRAN 90/95 programming language.

Learning Outcomes This is a basic introduction to modern Fortran. At the end of the course, students should be able to write significant programs in Fortran, and to be able to start working on existing programs written in modern Fortran (i.e. in the Fortran 90/95 style). It will not cover obsolete features of Fortran, some of the more advanced aspects, or most of the extensions introduced by Fortran 2003.

Prerequisites No Required No

Course Content During the course, some of the main issues concern Fortran will be taught, such as • computers and binary system. • Hardware and software. • Program development cycle, algorithms and flow diagrams. • Alphabet and syntax of FORTRAN. • Operators. • Selection structures and loops. • Arrays. Functions and subroutines. • Recursion. • Formatted input-output. • Files. • Dynamic data.

Teaching Methodology Lectures (3 hours weekly), Laboratory sessions (2 hour weekly) and Recitation (1 hour weekly).

Bibliography 1. Ματαράς Δ. και Κουτελιέρης Φ., ‘Προγραμματισμός Fortran 90/95 για επιστήμονες και μηχανικούς’, Εκδόσεις ΤΖΙΟΛΑ, 1η, 2003, ISBN 960-8050-43-Χ 2. Nyhoff L. and Leestma S., ‘Introduction to Fortran 90’, Prentice Hall 1999, ISBN 0-13-013146-6.

3. Καράκος Αλέξανδρος, 2008, Fortran 77/90/95 και Fortran 2003, Εκδόσεις Κλειδάριθμος.

Assessment Final exam, midterm exam, homework (theoretical and programming assignments) and quizzes. Labs + home exercises 20%

Midterm 20% Final Exam 60%

The total grade will be calculated based on the above percentage table.

Language Greek

http://www.tziola.gr/e-shop/author_details.asp?author_id=157

http://www.tziola.gr/e-shop/author_details.asp?author_id=158

http://www.tziola.gr/e-shop/publisher_details.asp?publisher_id=1

Course Title Physics for Mathematicians Course Code PHY103 Course Type Compulsory for students in Applied and Pure Mathematics, Elective for the rest. Level Undergraduate Year / Semester 2017 / Fall Teacher’s Name

ECTS 6 Lectures / week 2 X 1.5h Laboratories / week

0


To provide a broad and fairly coherent understanding of Modern Physics through methods of certain mathematical sophistication (undergrad level), and to develop students’ ability to actually apply these methods in order to obtain quantitative results (in closed analytical form) in particular physical situations

Learning Outcomes The students will learn what the most fundamental secrets of Nature are, both at the “classical” and “quantum” level (together with currently existing uncertainties on certain issues/questions that still remain open), and will be trained on how to practically use mathematical methods and apply them to particular physical situations in order to obtain (analytical) results

Prerequisites Officially none Required Officially none

Course Content Elements of Lagrangian and Hamiltonian Mechanics (and reference to Hamilton-Jacobi formulation as preparation for the passage to Quantum Mechanics). Elements of Electromagnetism/Classical Electrodynamics (Maxwell-Lorentz theory) – Introduction to the Special Theory of Relativity. Elements of Quantum Mechanics: quantum states as vectors - and observables as (self-adjoint) operators - in Hilbert spaces, position and momentum representations and Fourier transforms, physical meaning of eigenvalues and eigenstates of Hermitian operators, solution of Schrödinger equation (viewed as an ordinary or partial differential equation) in simple quantum systems – Uncertainty Principle – Ehrenfest and Hellmann-Feynman theorems – Symmetries and Generators, gauge symmetry (and some of its nontrivial consequences)

Teaching Methodology Lectures, in parallel with extensive (analytical) problem solving Bibliography 1. M. Levi, “Classical Mechanics with Calculus of Variations and Optimal Control - An Intuitive

Introduction” (2017), 2. L.N. Hand & J.D. Finch, “Analytical Mechanics” (2008), 3. T.A. Garrity, “Electricity and Magnetism for Mathematicians” (2015), 4. D.J. Griffiths, “Introduction to Electrodynamics” (2013), 5. D.J. Griffiths, “Introduction to Quantum Mechanics” (2005), as well as instructor’s notes influenced by: L.D. Faddeev & O.A. Yakubovskii (“Lectures on Quantum Mechanics for Mathematics Students” (2009)), I. Dolgachev (“A Brief Introduction to Physics for Mathematicians” (1996)), B.C. Hall (“Quantum Theory for Mathematicians” (2013)), J. M. Rabin (“Introduction to Quantum Field Theory for Mathematicians” (1995)), G.B. Folland (“Quantum Field Theory - A Tourist Guide for Mathematicians” (2008)) (and adjusted to the right level).

Assessment 20% Project, 40% Midterm Exam, 40% Final Exam Language Greek

Course Title Calculus Ι · Bibliography S.Ntoyas (in Greek) Calculus . Ι, Leader books. M.Spivak...

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