1. Calculus Ι (331-1002)[-C-]
Semester A Course (4 hours Theory and 1 hour Lab/Tutorial Exercises)
5 Teaching Points and 6 ECTS Points
Course Contents: :
The set of the real numbers, natural numbers and the principle of Mathematical induction and the well-ordering principle, the principle of completeness. Sequences and convergence of sequences. Functions, continuity, derivatives, fundamental theorems of calculus. De l’Hôpital’s rule. Taylor’s theorem. Introduction to integration. Indefinite and definite integrals, calculation of integrals, mean value theorem.
Prerequisite(s):
School’s Mathematical Analysis.
Follow-up Courses:
Calculus ΙΙ, Calculus ΙΙΙ, Real Analysis, Ordinary Differential Equations, Numerical Analysis.
Textbooks used:
1. M. Spivak, Calculus. University Publications of Crete, 1991.
2.
S.Κ. Pichoridis, Calculus Ι, Sychroni Epochi Publ. Co., Athens, 1996.
2. Calculus ΙI (331-2002)[-C-]
Semester B Course (4 hours Theory and 1 hour Lab/Tutorial Exercises)
5 Teaching Points and 6 ECTS Points
Course Contents: :
Series of numbers. Power series. Indefinite integrals. Definite integrals, the Riemann integral. Improper integrals. Introduction to the Laplace transformation.
Prerequisite(s):
Calculus I (331-1002) [-C-].
Follow-up Courses:
Calculus ΙΙΙ, Real Analysis, Ordinary Differential Equations, Numerical Analysis.
Textbooks used:
1. M. Spivak, Calculus. University Publications of Crete, 1991.
2. D. Georgiou, S.Iliadis, Th. Megaritis, Real Analysis, Patras, 2010.
3. S.Κ. Pichoridis, Calculus Ι, Sychroni Epochi Publ. Co., Athens, 1996.
4.
R.L. Finney, M.D. Weir, F.R. Giordano, Thomas Calculus, Volume II, University Publications of Crete, 2012.
3. Applied Linear Algebra I (331-1154) [-C-]
Semester A Course (3 hours Theory and 1 hour Lab/Tutorial Exercises)
4 Teaching Points and 5 ECTS Points
Course Contents: :
Linear equations and systems of linear equations, matrices and algebra of matrices, transpose of a matrix, square matrices, inverse of a matrix, symmetric, antisymmetric, and orthogonal matrices, similar matrices, block matrices, rank of a matrix, trace of a matrix. Determinants and their properties, Cramer’s theorem, adjoint of a matrix and computation of inverse using the adjoint, the space Rn, polynomials of matrices, characteristic polynomial, eigenvalues, eigenvectors, Cayley-Hamilton theorem, minimum polynomial.
Prerequisite(s):
School Algebra.
Follow-up Courses:
Applied Linear Algebra II (331-1156). Ordinary Differential Equations (331-2351). Numerical analysis and programming (331-2654).
Textbooks used:
1. M. A. O. Morris, Introduction to Linear Algebra, Pneumaticos Publ., 1980.
2. G. Strang, Linear Algebra and Applications, University Publications of Crete, 2001.
3.
G. Donatos, M. Adam, Linear Algebra. Theory and Applications, Gutenberg, Athens, 2008.
4. Applied Linear Algebra II (331-1159) [-C-]
Semester Β Course ( 2 hours Theory and 2 hours Lab/Tutorial Exercises)
4 Teaching Points and 6 ECTS Points
Course Contents: :
Vector spaces and subspaces. Linear combinations, finitely generated subspaces. Row space of a matrix. Linear dependence, basis and dimension. Dimension and subspaces. Linear transformations and applications in systems of linear equations. Matrix representation of a transformation. Change of basis matrix. Matrices and linear transformations. Matrix polynomials. Matrix diagonalization and eigenvectors. Jordan canonical form. Spaces with inner product, Cauchy-Schwarz inequality, orthogonality and orthonormal sets of vectors, Gram-Schmidt orthogonalization process. Quadratic forms.
Prerequisite(s):
Applied Linear Algebra I (331-1155).
Follow-up Courses:
Textbooks used:
1. A. O. Morris, Introduction to Linear Algebra, Pneumaticos Publ., 1980..
2. G. Strang, Linear Algebra and Applications, University Publications of Crete, 2001.
3.
G. Donatos, M. Adam, Linear Algebra. Theory and Applications, Gutenberg, Athens, 2008.
5. Real Analysis(331-2603) [-C-]
Semester D Course (3 hours Theory and 1 hour Tutorial Exercises)
4 Teaching Points and 6 ECTS Points
Course Contents: :
Uniform continuity, sequences of functions, pointwise convergence and uniform convergence of sequences of functions, series of functions, introduction to the Riemann-Stieltjes integral.
Prerequisite(s):
Calculus I and II (331-1001 and 331-2001).
Follow-up Courses:
Special Topics in Real Analysis (331-3871). Elements of Measure Theory (331-4921).
Textbooks used:
Walter Rudin, Principles of Mathematical Analysis, Leader Books Publ., Athens, 2000.
6. Discrete Mathematics (331-8141)[-E-]
Semester H Course (2 hours Theory and 1 hour Tutorial Exercises)
3 Teaching Points and 5 ECTS Points
Course Contents: :
Graphs and categories of graphs, paths, circuits and cycles, Eulerian circuit, Euler-Hierholzer theorem, the Königsberg bridge problem, Fleury’s algorithm. Hamiltonian cycles and the traveling salesman problem. Ore’s and Dirac’s theorems. Dijkstra’s algorithm, nearest neighbor algorithm. Representation of graphs, isomorphisms between graphs, planar graphs, Euler’s formula for connected and planar graphs. Colorings of graphs, Heawood’s theorem. Trees and binary trees, spanning and minimum spanning trees. “Depth first search” technique, Prim’s and Kruskal’s algorithms.
Prerequisite(s):
Elementary knowledge of Combinatorics.
Textbooks used:
C. L. Liu, Elements of Discrete Mathematics, University publications of Crete, 1994.
7. Special Topics in Real Analysis (331-3871)[-E-]
Semester E Course (2 hours Theory and 1 hour Tutorial Exercises)
3 Teaching Points and 5 ECTS Points
Course Contents: :
The notion of the metric and of the metric space, functions in metric spaces, continuity and uniform continuity, isometries and homeomorphisms, topology of metric spaces, complete metric spaces, totally bounded metric spaces, compact metric spaces, separable metric spaces, connected metric spaces, path connectedness.
Prerequisite(s):
Real Analysis (331-2603).
Textbooks used:
Robert C. Werde, Murray Spiegel, Advanced Mathematics Schaum’s series, Tziola Publ., Thessaloniki. Walter Rudin, Principles of Mathematical Analysis, Leader Books Publ., Athens, 2000. P. Tsamatos, Topology, Tziola Publ., Thessaloniki, 2009.
8. Numerical Analysis and Programming(331-2654)[-C-]
Semester E Course (3 hours Theory and 2 hours Laboratories)
5 Teaching Points and 5 ECTS Points
Course Contents: :
Gauss elimination. LU and Choleski factorization. Stability of linear systems. General relaxation methods. Methods of Jacobi and Gauss-Seidel. The power method for the numerical calculation of the dominant eigenvalue and dominant eigenvector. Langrange, Hermite and spline interpolation. Chebyshev polynomials. Weierstrass’ theorem. Numerical integration. Root finding. Solution of non-linear systems. Numerical solution of ordinary differential equations. Taylor and Runge-Kutta methods. Introduction to programming with C and C++. Numerical methods in C and C++.
Prerequisite(s):
Calculus I, II and III. Applied Linear Linear Algebra I and II. Real Analysis and Ordinary Differential Equations.
Textbooks used:
1) Michael Ν. Vrachatis, Numerical Analysis, Publ. Ellinika Grammata, Athens, 2002. 2) Nicolaos Misirlis. Numerical Analysis. An algorithmic approach. Publ. Nicolaos Misirlis 2009. 3) C. E. Lazos, C++. Theory and practice, 2nd Edition, Thessaloniki, 2004.
Course material
9. Calculus III(331-2252) [-C-]
10. Introduction to Probabilities and Combinatorics (331-1203) [-C-]
11. Applied Analysis (331-2605) [-SC-]
12. Measure Theory (331-4922) [-E-]
13. Measure Theory (Postgraduate Course)
14. Measure and Probability (Postgraduate Course)
15. Introduction to Combinatorics and Probabilities(331-1205) [-C-]