Deterministic Methods of Optimization

Introductory concepts: Convex and Concave functions. Solving NLPs with one variable. Iterative methods of finding extrema of functions in Rn, n>1.

Matrix Theory

Prerequisite matrix theory - Matrix polynomials and normal forms - Functions of matrices - Inner products and matrix norms - Normal matrices - polar decomposition - singular value decomposition - Kronecker and Hadamard products - Nonnegative matrices - Generalized inverses.

Computational Mathematics

Interpolation and approximation with piecewise polynomials and splines , Numerical linear algebra: Gauss elimination for linear systems pivoting, LU- factorization and an introduction to the stability of systems and algorithms, norms of vectors and matrices, condition number, Iterative methods, Introduction to the numerical solution of eigenvalue-eigenvector problem, Numerical solution of ODEs (existence and uniqueness of initial value problem). Euler method, Taylor method, Runge-Kutta methods and multistep methods.

Κλασική Διαφορική Γεωμετρία ΙΙ

Στοιχεία διαφορικών μορφών – Η μέθοδος του κινουμένου τριάκμου (Θεμελιώδεις εξισώσεις της θεωρίας επιφανειών. Αναλλοίωτες μορφές. Σφαιρική απεικόνιση. Το τρίακμο Darboux. Κάθετη καμπυλότητα, γεωδαισιακή καμπυλότητα, γεωδαισιακή στρέψη. Πρωτεύουσες καμπυλότητες) – Εσωτερική Γεωμετρία των επιφανειών.

Linear Geometry

Multidimensional affine spaces – Affine subspaces – Affine mappings.

Functional Analysis

Basic notions - Metric spaces - Normed spaces - Inner product spaces - Linear operators and functionals - Norms in B(X,Y), Hahn-Banach, Banach-Steinhaus, open mapping and closed graph Theorems.

Measure Theory

Lebesgue measure on the real line - Measurable functions - Lebesgue integral - Monotone and dominated convergence theorems - Comparison of integrals of Riemann and Lebesgue - The fundamental theorem of Calculus for Lebesgue integral - Abstract measure theory - Signed and complex measures - Product measures - Fubini’s theorem.

Continuum Mechanics

Introduction to Tensor Analysis - Lagrangian and Eulerian description of the motion - Local and total derivatives - Streamlines and pathlines of particles - Potential flow - Strain tensor - Displacement vector - Rate of deformation tensor - Velocity distribution in infinitesimal regions - Circulation and turbulent flow - The equation of continuity - Mass forces, stress vector and stress tensor - Equations of motion of a continuum - Ideal and Newtonian fluids - Euler and Navier-Stokes equations - Applications - examples.


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