Search results for “композиции моделей”

What geometries there are and what they have in common — Klein's Erlangen programme; the school 3–4–5 triangle and the ellipsoid; the cube and the brick as finite geometries; metric spaces; groups, quantifiers in the axioms; the group of bijections and the order of composition; left and right actions of a group on a set; the metrics $d_1$, $d_2$, $d_\infty$ on $\mathbb{R}^n$; isometries, subgroups; the order of an element; orbits, stabilisers, transitive actions; the list of geometries in the course — the Euclidean plane, the regular triangle and $S_3$, the cube and its 48 symmetries, the Poincaré model of the Lobachevsky plane; fundamental domains; subgeometries — the square in the cube.

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.

Why homology is needed — the fundamental group as a functor, homotopy groups, bordism and its combinatorial approximation; chain complexes and their homology; abstract simplicial complexes, geometric realisation and triangulations; the complex of simplicial chains, the proof of $d^2 = 0$, the simplicial homology of a segment, of two points and of the boundary of a triangle; reduced homology.