Search results for “структурная минимизация риска”

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Topology 2. Lecture 2

2026LectureФёдор Вылегжанин

Where simplicial homology shows up in real life — topological data analysis. The union of balls around a point cloud, persistence modules and their decomposition into a sum of interval modules over a field (the structure theorem and the algorithm), the barcode; the Vietoris–Rips and Čech complexes; the stability theorem for barcodes, the Hausdorff, Gromov–Hausdorff and Wasserstein distances; open covers and their nerves, Leray covers and the nerve theorem, the category of covers, paracompactness, Lebesgue covering dimension, Čech homology and cohomology.

520

Algebra 1. Lecture 1

2026LectureАртём Авилов

What algebra studies — algebraic structures. Groups (definition, examples, additive notation, uniqueness of the identity and of inverses), rings (variants of the axioms, rings with and without identity), fields and division rings, vector spaces and modules, algebras over a field. The symmetric group $S_n$ — permutations, non-commutativity, two-row notation, cycles and transpositions, decomposition into transpositions, disjoint permutations commute, decomposition into disjoint cycles and its uniqueness, powers, inversions, parity and the sign of a permutation, multiplying by a transposition changes the sign.

310