COURSE DESCRIPTIONS
ΜAS101 - Calculus Ι
Properties of real numbers. The basic properties of supA, infA. Sequences of real numbers, limits of sequences. Real valued functions, the inverse of a function, limits of functions, continuous functions, uniform continuity, the Intermediate Value Theorem, the Extreme Value Theory. Derivatives, graphs of functions, the Mean value theorem, L’Hopital’s rule.
ΜAS102 - Calculus II
Riemann integral, integrability of continuous, monotone functions. The fundamental Theorem of calculus. Areas of regions in the plane, volumes of solids of revolution. The indefinite integral, integration by parts, integration by change of variables, integration of rational functions. Taylor’s formula. Infinite series, tests of convergence, absolutely convergent series, conditionally convergent series, Leibniz’s Theorem, product of series.
MAS121 - Linear Algebra I
Numbers, equivalence relations. Groups, Examples (symmetric, cyclic, dihedral). Isomorphism. Rings and Fields. Examples. Vector spaces, basis, dimension. Linear maps. Matrices and linear maps. Rank, change of basis matrix. Determinant. Linear systems.
MAS122 - Linear Algebra II
Polynomial Ring. Eigenvalues, eigenvectors. Diagonalization and applications. Theorem of Cayley – Hamilton, minimal polynomial. Generalized eigenspaces, nilpotent endomorphisms Jordan canonical form. Inner product spaces (Gram – Schmidt). Orthogonal, self dual endomorphisms. Bilinear, quadratic forms.
MAS131 – Basic Mathematics
Methods and applications of differentiation. Methods of integration and applications. Improper Integrals. Power series. Fourier series. Elements of analytic geometry on the place and in space. Functions and surfaces. Polar coordinates. Partial derivatives and Lagrange multipliers. Multiple integration and Jacobien.
MAS191 - Mathematics with computers
Preliminaries: Basic Matlab commands. Matlab as a programming language. Real and complex numbers, vectors, matrices. Representation of numbers, vectors& matrices. Simple Matlab programs. Matrices: General notions. Matrix operations with Matlab. Computation of determinants and inverses. Eigenvalues and Eigenvectors: General notions of Eigenvalues and Eigenvectors. Computation of them with Matlab. Special emphasis on the complex case. Diagonal table matrices. Plots with Matlab: Simple plot, two- and three- dimensional plots. Special plots: Phase planes, contour plots, flows. Linear on OPES. Special topics on differential equations. Multivariate calculus. Fast Fourier Transform.
MAS201 - Multivariate Differential calculus
Spaces with norm (examples, n- dimensional Euclidean space, equivalent norms, Cauchy – Schwarz inequality).
Open, closed sets, limits, continuity. Compactness (Theorem of Heine – Borel, Bolzano – Weierstrass). Vector valued functions of one real variable. Partial derivates. Total differential. Mean value theorem, Taylor’s Theorem. Implicit and inverse function theorems. Lagrange multipliers.
MAS202 - Multivariate Integral Calculus
Integration of continuous functions with compact support. Transformation theorem. Integrable functions and sets, properties. Volumes. Theorem of Fubini. Convergence theorems. Transformation Theorem, applications. Parameterized surfaces, partition of unity. Surface and curve – integrals. Differential forms. Theorem of Stokes, applications.
MAS203 - Ordinary Differential Equations
Basic notions. Solutions techniques for first – order equations& physical applications. Theorems of Existence and Uniqueness. Linear systems& exponential of matrices. Higher order linear equations. Method of power series: Smooth and singular solutions. Smooth dependence of solutions on parameters.
MAS 223 - Number Theory
Divisibility theory in the integers. The Euclidean algorithm. Primes and their distribution. The fundamental theorem of arithmetic The theory of consequences. Fermat´ s little theorem. The quadratic reciprocity law. Perfect numbers. Representation of integers as sums of squares. Fibonacci numbers, Continued fractions.
Pell´ s equation.
MAS 251 – Probability I
Probability, random variables, distribution functions, independence, expected value, moment generating functions, modes of convergence of sequences of random variables, laws of large numbers, introduction to central limit theorems.
MAS252 - Statistics Ι
Statistics. Sufficiency and completeness. Exponential families of distributions. Unbiasedness, unbiased estimators. Cramer – Rao inequality. Method of moments, maximum likelihood estimators. Asymptotic properties of estimators. Bayes estimation. Introduction to confidence intervals and to hypothesis testing problems.
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MAS271 - Numerical Analysis I
Propagation and estimation of errors: Floating-point arithmetic - Rounding error analysis - Loss of significance - Stability and condition of problems and algorithms - The symbolism - Richardson extrapolation. The solution of nonlinear equations: Fixed-point iteration - Order of convergence and asymptotic error constant - The Newton and the secant methods - Multiple roots - Always convergent methods (the bisection and the regula falsi methods) - Aitken´s acceleration process. Solution of linear systems: Direct methods (Gauss elimination and LU-decomposition) - The need for partial pivoting and for scaling - Cholesky´s method for symmetric and positive definite systems - The computation of the determinant and the inverse of an nxn matrix - the least squares method for over-determined systems. Interpolation and quadrature: Lagrange interpolation (Existence and uniqueness - Cardinal and Newton representations of the interpolating polynomial - The error of the interpolating polynomial) - Hermite interpolation (Existence and uniqueness - Cardinal representation of the interpolating polynomial - The error of the interpolating polynomial) - Newton-Cotes quadrature rules - The precision of a quadrature rule - Detailed description and analysis of the trapezoidal and the Simpson rules - Composite rules.
MAS301 – Real Analysis
The real number system R, the least upper bound property and its consequences. Countable and uncountable sets. The Cantor ternary set. Introductory theory of metric spaces. The metric spaces R and Rn . Compact sets. Heine – Borel Theorem, Bolzano – Weierstrass Theorem. Sequences of real numbers, limit superior and inferior of a sequence. Cauchy sequences, series of real numbers. Complete metric spaces, Cantor intersection Theorem, The fixed point Theorem and applications. Continuous functions. Topological characterization of continuity. Continuity and compactness. Uniform continuity, Lipschitz functions. Sequences and series of functions. Pointwise convergence, uniform convergence. Uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation. The space C ([ a,b]) , the topology of uniform convergence.
ΜAS302 – Complex Analysis Ι
Complex numbers, Basic complex functions, Cauchy- Riemann equations, holomorphic functions, harmonic functions. (Exponential, trigonometric and logarithmic functions). Contontintegration, Cauchy’s theorem, Cauchy’s integral formula. Morera’s theorem, Liouville’s theorem, the fundamental theorem of algebra. The Maximum modulus theorem. Taylor series, Laurent series, calculus of residues. Conformal mapping, linear fractional transformation.
ΜAS303 – Partial Differential Equations
Separation of variables – Fourier series. First order Partial Differential Equations. Nonlinear first order Partial Differential Equations. Linear second order Partial Differential Equations. Elliptic, Parabolic and Hyperbolic Partial Differential Equations.
ΜAS304 – Functional Analysis
Metric spaces: Examples and elements of the theory of metric spaces. Banach spaces: Norm, dimension and compactness, bounded operators, linear forms, dual space, the spaces LP, 1≤p≤∞, Hilbert spaces: Inner products, orthogonal sums, orthonormal bases, the Riesz representation theorem, the adjoint operator , self – adjoint, unitary and normal operators. Fundamental theorems for Banach spaces: the Hah – Banach theorem, reflexive spaces, the uniform boundedness theorem, weak and strong convergence, the open mapping and closed operators, closed graph theorem. Applicators: The fixed point theorem and its applications to the theory of linear, integral and differential equations, applications to the theory of approximation.
MAS321 - Introduction to Algebra
Basic properties of groups. Cayley´s theorem. Subgroup and Lagrange´s theorem. Normal subgroups and factor groups. First isomorphism theorem. Group actions. Basic properties of rings. Ideals. R – modules over Eucledean domains and the fundamental theorem of finitely generated abelian groups.
MAS331 - Classical Differential geometry
Curves in ( length, orientation, parameterization). Curves in (normal field, curvature, Frenet equations). Curves in (curvature, torsion, Frenet equations). Surfaces in : Parameterization, tangent space, first and second fundamental form, Curvature, geometric interpretation of curvature, examples. Inner geometry of surfaces (local isometry, Christoffel symbols, Theorema Egregium, vector fields, parallel transport, geodesics). Theorem of Gauss – Bonnet.
MAS350 – Stochastic Processes
Basic concepts, continuous and discrete time Markov processes, birth and death processes, Poisson processes, introduction to martingales, Brownian motion.
MAS351 – Probability II
Multivariate distributions, distribution of functions of random variables, conditional expected value and covariance, order statistics, distributions related to the normal distribution, characteristic functions, modes of convergence of a sequence of random variables, laws of large numbers, central limit theorem.
MAS352 – Statistics ΙΙ
Confidence intervals. Introduction to hypothesis testing problems. Neyman – Pearson Lemma, monotone likelihood ratio. Locally most powerful tests, uniformly most powerful tests. Asymptotic efficiency, uniformly most powerful unbiased tests. Likelihood ratio tests
MAS371 - Numerical Analysis II
Preliminaries: Basic definitions and theorems of Linear Algebra - Lagrange and Hermite interpolation - Newton-Cotes quadrature rules. Vector and matrix norms: Basic definitions and properties - Induced matrix norms - Perturbed linear systems (perturbation analysis) - Condition of linear systems - Iterative refinement. Methods for eigenproblems: The Gershgorin theorems - The Rayleigh quotient - The power and the inverse iteration methods - Similarity transformation methods (Givens and Householder for symmetric matrices - Basic forms of the LR and the QR algorithms) - Sturm sequence property for the eigenvalues of symmetric tri-diagonal matrices. Iterative methods for linear systems: General iterative method - The methods of Jacobi, Gauss-Seidel and SOR - Convergence theorems - Asymptotic rate of convergence - Introduction to the theory for the optimum SOR relaxation parameter. Orthogonal polynomials and Gauss quadrature rules: Zeros of orthogonal polynomials - Three-term recurrence relation - Legendre, Chebyshev, Laguere, Hermite and Jacobi polynomials - Gauss quadrature rules (Legendre, Chebyshev, Laguere, Hermite and Jacobi.)
ΜAS401 – Measure Theory and Integration
General revision: Sets, orderings, cardinality, metric spaces. Measures: Algebras and σ- algebras, additive and σ- additive measures, outer measures, Borel measures on the real line. Integration: measurable functions, integration of positive functions, integration of complex valued functions, 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.
ΜAS402 – Complex Analysis ΙΙ
Compactness and convergence in the space of analytic functions. The space of meromorphic functions. Riemann mapping theorem. Weierstrass factorisation theorem. Analytic continuation (Schwarz reflection principle, Monodromy theorem). Entire functions. Elements of Geometric theory.
MAS403 – Stability of Dynamical Systems
Asymptotic behaviour of non-linear systems of ordinary differential equations: stability. Perturbation theory of systems having periodic solutions. Perturbations of two-dimensional autonomous systems. Poincaré-Bendixson theory.
ΜAS418 – Introduction to Fourier Analysis
Inner products, Hilbert spaces, orthogonal systems, completeness, periodic functions, trigonometric polynomials, Fourier series, pointwise convergence, the Dirichlet theorem, Gibb´s phenomenon, Parseval theorem, Cesàro and Abel summability the Fejer and Poisson theorems, the Riemann – Lebesgue Lemma, convergence of special trigonometric series, the local Riemann theorem. Differentiation and integration of Fourier series, Fourier transform, Plancherd´s formula, convolution, applications to Partial Differential Equations.
ΜAS419 – Topics in Analysis
Topics from real analysis, complex analysis, harmonic analysis or differential equations.
MAS422 - Introduction to Coding Theory
Introduction to finite fields. Vector spaces over finite fields. Linear codes. Encoding and decoding with a linear code. Syndrome decoding. Hamming codes. Cyclic codes
MAS424 - Theory of Rings and Modules
Rings and ideals. Homomorphism theorems. Unique factorization domains and principal ideal domains. Factor rings. Prime and maximal ideals. R – modules and homomorphisms. Finitely generated R- modules. Noetherian rings.
MAS425 - Theory of Groups
Generators and relations. Homomorphism theorems. Direct and semidirect products. Group actions. Sylow theorems and p – groups. Simple groups. Composition series and the Jordan – Hőlder theorem. Soluble and nilpotent groups.
MAS426 - Group Representation Theory
Semisimple rings, irreducible R – modules. Splitting fields. Relation between characters and representations. Frobenius theorem. Representations of the symmetric group.
MAS427 - Galois Theory
Polynomial rings, irreducible polynomials. Field extensions and splitting fields. Automorphisms and fixed fields. Normal extensions and Galois extensions. The fundamental theorem of Galois theory. Solution by radicals.
ΜAS429 – Topics in Algebra
Topics from Algebra.
MAS431 - Introduction to differentiable manifolds.
Manifolds, Tangent space. Partition of unity. Theorem of Sard. Vector fields, flows. Frobenius Theorem. Differenital forms. Theorem of Stokes. De Rham Theorem.
MAS432 - Introduction to Riemannian Geometry
Riemannian manifolds. Geodesics, exponential map, normal coordinates. Gauss lemma. Theorem of Hopf – Rinow. Curvature. Jacobi fields. Theorem of Bonnet – Myers, Synge – Weinstein and Hadamard – Cartan.
MAS433 - Introduction to algebraic topology
Topological spaces. Continuous functions. Separation Axioms. Compact, connected sets. Homotopy, Fundamental group, covering spaces. Introduction to Homology.
MAS434 - Algebraic Topology
Homology theory and applications. Cohomology. Universal coefficient theorem. Products. Kuenneth formula. Thom isomorphism. Poincare duality.
ΜAS439 – Topics in Geometry
Topics from differential geometry, algebraic geometry and algebraic topology.
MAS451 - Linear Models I
The Simple Linear Regression Model: Estimation, Confidence Intervals, Hypothesis Testing. The Multiple Linear Regression Model: Estimation, Confidence Intervals, Hypothesis Testing. Model Adequacy and Model Selection. Polynomial Regression.
MAS452 - Linear Models II
Analysis of variance with one or more fixed-effects, Analysis of variance with one or more random-effects, Analysis of covariance, Generalized linear models: estimation in (for example) logistic or logarithmic regression, asymptotic properties.
ΜAS454 – Nonparametric Statistics
Order statistics and their distributions. Sign tests, rank tests, confidence intervals, tolerance regions. Rank correlation coefficient and tests, linear regression. Kolmogorov-Smirnov tests, Lilliefors test. Contingency tables, X2 tests for goodness of fit, independence and homogeneity.
ΜAS455 – Sampling Theory
Survey design. Simple random sampling, stratified, systematic, cluster and multi-stage sampling. Mean and variance estimation, ratio estimators, regression estimators. Determination of optimum sample size. Sampling errors.
ΜAS456 – Time Series
Stationary processes, second order moments. ARMA and ARIMA processes. Maximum likelihood estimation, least squares estimators, Yule-Walker estimators. Prediction of stationary processes. Introduction to model selection.
ΜAS458 – Statistical Data Analysis
Exploratory statistics. Linear models and applications. Analysis of variance, classification analysis, data structure analysis, exploratory methods. Generalised linear models. Nonlinear models, robust methods, experimental design methods. Statistical computing methods and software. Biometric, econometric and other applications.
ΜAS459 – Multivariate Analysis
Multivariate Normal distribution, estimation of the mean vector and the covariance matrix, maximum likelihood estimation. Correlation coefficient, partial correlation coefficient and their distribution. T2- statistic and its distribution, T2- tests. Distribution of the sample covariance matrix, Wishart distribution, Principal components, canonical correlations, cluster and discriminant analysis. Introduction to multivariate analysis of variance: parameter estimation and tests.
MAS466 - Survival Analysis
Censored data, truncated data. Survival function and hazard function. Nonparametric estimation of the survival function and the hazard function. Parametric models for the hazard function. Counting processes and martingales. Semiparametric Cox model. Tests for one or more populations, tests of class – K.
MAS468 – Topics in Probability
Topics from probability.
MAS469 – Topics in Statistics
Topics from statistics.
MAS471 - Numerical solution of ordinary differential equations
Numerical solution of ordinary differential equations: Linear multistep methods-theory and applications, Runge-Kutta methods, first order systems and stiffness, two-point boundary value problems.
ΜΑΣ472 – Numerical solution of partial differential equations
First and second order hyperbolic PDEs, the method of characteristics, finite difference techniques, the finite element method. Parabolic PDEs, methods for the solution of the one- and two-dimensional heat equation. Elliptic PDEs, finite difference methods for Poisson's equation.
MAS473 – Finite Element Method
Variational formulation of boundary value problems. Galerkin Method. Basis functions and discretization. Stiffness matrix and methods of solving linear systems. Error estimates. Collocation method, least squares method, and Rayleigh-Ritz method. Finite element methods for parabolic equations.
ΜAS481 – Applied Mathematical Analysis
Calculus of variations. Laplace transform. Fourier analysis. Special functions. Integral equations. Asymptotic analysis.
ΜAS482 – Classical Mechanics
Newton’s Laws. Central Forces. Moving Coordinate Systems. Systems of Particles. Motion of Rigid Bodies. Language’s Equations.
ΜAS483 – Fluid Mechanics
Coordinate systems. Vector and tensor calculus. Surfaces and integral theorems. Conservation laws. Narier-Strokes equations. Partial differential equations and methods of solution. Flows with analytical solution. Flow Potential Theory and related problems.
MAS484 – Introduction to mathematical modelling
This course emphasizes the role of mathematical modelling as a tool for learning and appreciating mathematical techniques. Applications are drawn from diverse areas such as discrete dynamical systems, graphs and networks, linear programming, transportation. Extensive use of computer software is made throughout the course.
MAS499 – Independent Study
An independent study with sufficient elements of initiative and novelty under the guidance of a faculty member.