Title  ΣΕΜΙΝΑΡΙΟ ΠΡΟΒΛΗΜΑΤΩΝ ΙΙ / SEMINARIO PROVLĪMATŌN II 
Code  0148 
Faculty  Sciences 
School  Mathematics 
Cycle / Level  1st / Undergraduate 
Teaching Period  Spring 
Coordinator  Romanos diogenis Malikiosis 
Common  No 
Status  Active 
Course ID  600017209 
Programme of Study: UPS of School of Mathematics (2014today)
Registered students: 12
Orientation  Attendance Type  Semester  Year  ECTS 

Core  Elective Courses  2  1  2 
Academic Year  2019 – 2020 
Class Period  Spring 
Faculty Instructors 

Weekly Hours  2 
Class ID  600147675

Type of the Course
 Scientific Area
 Skills Development
Course Category
Knowledge Deepening / Consolidation
Mode of Delivery
 Face to face
Digital Course Content
Language of Instruction
 Greek (Instruction)
 English (Instruction, Examination)
General Competences
 Work in an international context
 Advance free, creative and causative thinking
Course Content (Syllabus)
Analysis
1. Real and complex numbers.
2. Sequences and series of numbers.
3. Functions of one real variable: continuity, differentiability, Taylor formula, Riemann integral.
4. Sequences and series of functions: pointwise and uniform convergence; differentiability and integrability term by term.
5. Power series, elementary functions.
6. Improper Riemann integral, functions defined by integrals (Euler integrals).
7. Solution of ordinary differential equations
8. Multivariate functions. FubiniTonelli theorem. Theorems of Green, Stokes, Gauss.
9. Lebesgue integral. Monotone and dominated convergence theorem.
Algebra and Geometry
1. General notions about some algebraic structures: groups, rings, fields.
2. General properties about polynomials with real and complex coefficients.
3. Finite dimensional vector spaces over real and complex numbers: base and dimension.
4. Linear transformations and matrices; eigenvalues, eigenvectors, diagonal form and applications.
5. Quadratic forms. Plane and and solid analytical geometry: lines, planes, conics, quadrics.
Number Theory
1. Divisibility, congruences modn.
2. Theorems of Fermat, Euler, Wilson.
3. Quadratic residues. Multiplicative structure of reduced residues modn.
Probability and Combinatorics
1. Random walks on the plane and space.
2. Geometric probability.
3. Generating functions.
Educational Material Types
 Notes
 Book
Course Organization
Activities  Workload  ECTS  Individual  Teamwork  Erasmus 

Lectures  26  0.9  
Reading Assigment  29  1.0  
Exams  5  0.2  
Total  60  2 
Student Assessment
Student Assessment methods
 Written Exam with Problem Solving (Summative)
Bibliography
Course Bibliography (Eudoxus)
1. Problems in Real Analysis: Advanced Calcuclus on the Real Axis, by T.L. Radulescu, V. Radulescu, T. Andreescu. Springer, 2009.
2. Putnam and Beyond, by R. Gelca, T. Andreescu. Second edition, Springer 2017.
3. Essential Linear Algebra with Applications: A Problem Solving Approach, by T. Andreescu. Springer 2014.
Last Update
17042020