Search results for “topoi”

Why the definition of a filtered category insists on non-emptiness: the empty colimit fails to commute with the empty limit, and that is the only failure.

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.

Singular homology — standard simplices and the inclusions of facets, singular simplices and chains, the differential; zeroth homology and path components, the homology of a point, reduced homology; chain maps and maps of homology, functoriality, the chain map induced by a continuous map; chain homotopies and the prism; the homotopy invariance theorem with a full proof via the prism operator; subcomplexes, quotient complexes and the homology of a pair.

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.

The Millennium Meeting, Paris, 24 May 2000, at which the seven problems were announced.

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.

Тематическое моделирование. Лекция курса «Машинное обучение» К. В. Воронцова. Задача тематического моделирования коллекции текстовых документов. Метод максимума правдоподобия.

Поиск ассоциативных правил. Лекция курса «Машинное обучение» К. В. Воронцова. Понятие ассоциативного правила и его связь с понятием логической закономерности.

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