Polynomials: divisibility, Euclidean algorithm, roots, cubic equation, fundamental theorem of algebra, Sturm theorem.

Geometry of finite dimensional Euclidean spaces: isometries, translation group, and rotation group.

Metric spaces: examples, open and closed sets, cluster points, compactness, connectedness, completeness. Banach fixed-point theorem. Relation to the geometry of Euclidean spaces. Heine-Borel theorem.

# Introductory seminar B

J. Vrabec: Metrični prostori, DMFA-založništvo, Ljubljana, 1990.

Proseminar B complements courses Analysis 1 and Algebra 1 with some additional content and strengthens the understanding of these courses by concrete examples.

Knowledge and understanding: Understanding the behavior of polynomials, geometry of Euclidean spaces, and the concept of the metric space.

Application: Application in geometry, natural science and other field of science.

Reflection: Understanding of the theory from the applications.

Transferable skills: The ability to design the problem, select an appropriate method, solve the problem, and analyse the results on test cases. The ability to formulate a problem in mathematical language. Skills in using the literature.

Lectures, exercises, homework, consultations

2 midterm exams instead of written exam, written exam

Oral exam

grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Matej Brešar:

BAHTURIN, Jurij Aleksandrovič, BREŠAR, Matej, ŠPENKO, Špela. Lie superautomorphisms on associative algebras, II. Algebras and representation theory, ISSN 1386-923X, 2012, vol. 15, no 3, str. 507-525. [COBISS-SI-ID 16299353]

BREŠAR, Matej. A unified approach to the structure theory of PI-rings and GPI-rings. Serdica mathematical journal, ISSN 1310-6600, 2012, vol. 38, no 1-3, str. 199-210. [COBISS-SI-ID 16355673]

BREŠAR, Matej. Near-derivations in Lie algebras. Journal of algebra, ISSN 0021-8693, 2008, vol. 320, no. 10, str. 3765-3772. [COBISS-SI-ID 14945113]

Emil Žagar:

JAKLIČ, Gašper, KOZAK, Jernej, KRAJNC, Marjetka, VITRIH, Vito, ŽAGAR, Emil. Hermite geometric interpolation by rational Bézier spatial curves. SIAM journal on numerical analysis, ISSN 0036-1429, 2012, vol. 50, no. 5, str. 2695-2715. [COBISS-SI-ID 16449369]

JAKLIČ, Gašper, VITRIH, Vito, ŽAGAR, Emil. Closed form formula for the number of restricted compositions. Bulletin of the Australian Mathematical Society, ISSN 0004-9727, 2010, vol. 81, iss. 2, str. 289-297. [COBISS-SI-ID 15540569]

KOZAK, Jernej, ŽAGAR, Emil. On geometric interpolation by polynomial curves. SIAM journal on numerical analysis, ISSN 0036-1429, 2004, vol. 42, no. 3, str. 953-967. [COBISS-SI-ID 13398617]