Search results for “k-средних”

Problems for the course "Topology 2" (F. E. Vylegzhanin, Independent University of Moscow, autumn 2026), sheet 1 of 9 September 2026 — simplicial homology and chain complexes. Zeroth homology and connected components; homology of a disjoint union and of a wedge; triangulations of the circle, the sphere and the Möbius band; the join and the product of simplicial complexes; the Euler characteristic and the Morse inequalities; the number of vertices of a triangulated surface; operations on chain complexes; exact sequences; the Smith normal form and the splitting of a complex.

Problems for the course "Topology 2" (F. E. Vylegzhanin, Independent University of Moscow, autumn 2026), sheet 2 of 16 September 2026 — singular homology and chain homotopies. A chain map from simplicial to singular chains; retractions and direct summands; the Hurewicz homomorphism from the fundamental group to the first homology; finite CW complexes and homology classes; homology of a union of a chain of spaces; homology of a simplex and of its skeleton; simplicial maps; the chain homotopy equivalence between a complex and its homology; chain homotopy as an equivalence relation; the five lemma.

Linear dependence and independence of the rows of a matrix, the criterion via a linear combination, and the connection to rank.

The axioms of a linear (vector) space, examples and counterexamples.

Linear dependence of vectors, basis and coordinates, dimension, subspaces, and linear span.

Линейные пространства над произвольным полем, подпространства, линейная зависимость, базис и размерность, сумма и пересечение подпространств, замена базиса.

An example, due to Serre, of a smooth projective surface in positive characteristic whose Picard scheme fails to be reduced, worked out via the two governing dimension inequalities.

International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.

International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.

International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.

International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.

International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.

International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.

50th International Mathematical Olympiad. Bremen, Germany, 2009.

54th International Mathematical Olympiad. Colombia, 2013.

63rd International Mathematical Olympiad. Oslo, 2022.

64th International Mathematical Olympiad. Chiba, Japan, 2023.