Compulsory

Name:
Compulsory

Γενικά Μαθηματιικά Ι - Τμήμα Γεωλογίας

Πίνακες (πράξεις, ιδιότητες), ορίζουσες (ιδιότητες), αντίστροφος πίνακα, βαθμίδα πίνακα (rank), διερεύνηση και επίλυση γραμμικών συστημάτων. Αναλυτική Γεωμετρία (εξίσωση ευθείας και επιπέδου, εξίσωση κύκλου κλπ.) κυλινδρικές και κωνικές επιφάνειες. Συναρτήσεις μίας μεταβλητής. Σειρές Taylor-Maclaurin, γραφική παράσταση. Ολοκληρώματα. Συναρτήσεις πολλών μεταβλητών (όριο, συνέχεια κλπ.), διπλά και τριπλά ολοκληρώματα.

Complex Analysis

Complex numbers, the complex plane, topology of the plane, elementary complex functions - Holomorphic functions, Cauchy-Riemann equations - The complex integral, Cauchy's theorem and integral formula - The maximum principle, theorems of Morera and Liouville, the Schwarz lemma - Power series, the identity theorem. - Laurent series, singularities, residues.

Stochastic Strategies

Stochastic problems - Stochastic networks - Stochastic problems of tools replacement and repairing - Renewal theory - Inventory.

Probability Theory II

The algebra of events - Probability Space - The axioms of Probability - Random variables - The notion of stochastic distribution - Multidimensional random variables - Multidimensional distribution functions - Marginal distributions - Denumerable multidimensional random variables - Continuous multidimensional distributions - Multidimensional normal distribution - Stochastic independence - Conditional Probability - Conditional density - Conditional distributions - Mean values for multidimensional random variables - Conditional mean values - Regression line - Mean square error - Random variabl

Numerical Analysis

Structure of Computational systems and algorithms, number systems and errors - Interpolation and approximation (interpolation by Lagrange and Newton polynomials) - Numerical integration (midpoint, trapezoid and Simpson’s rules, Romberg integration) - Numerical solution of non-linear equations (bisection method, secant, regula-falsi and modified regula-falsi, Newton’s method) - Introduction to iterative methods for linear systems and ODE.

Classical Differential Geometry I

Definition of a curve - The method of moving frame - Fundamental Theorem of Curves Theory- Definition of a surface - Curves on surfaces - Fundamental forms - Asymptotic lines - Christoffel symbols - Theorema egregium - The Gauss mapping - Fundamental Theorem of Surface Theory

Introduction to Real Analysis

Real numbers - Countable and uncountable sets - Sequences and series - permutations of series - representations of real numbers - The Cantor set and Cantor’s function - Special classes of functions (monotone, bounded variation, absolutely continuous, convex) - Sequences and series of functions - uniform convergence and applications - nowhere differentiable continuous functions - space filling curves - equicartinuity - Azzela’s-Ascoli’s theorem - Weierstrass approximation theorem - Lebesque’s measure.

Mathematical Methods in Operational Research

What is a stochastic process - Queuing Theory: birth-death processes - some well-known queuing systems - Markov Chains: n-step transition probabilities - classification of states - steady-state probabilities and mean first passage times - absorbing chains.

Statistics

Elements of probability theory - Distributions of some useful statistics - Descriptive statistics - Methods of point estimation - Confidence intervals and tests of hypotheses for the mean, the variance and the proportion for one and two samples - Test of Goodness-of-Fit - Contingency tables - Tests of homogeneity - The method of least squares-Regression - Tests of hypotheses and Confidence intervals in simple linear Regression - Simple, multiple and partial correlation coefficient - Analysis of variance - The one-way layout - The two-way layout with and without interaction - Non-parametric

Differential Equations

Differential equations of first order - The method of Pickard - Linear differential equations of order n2 - Reduction of the order of a differential equation - Euler’s equations systems of differential equations.

Pages X