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Local ring - ... functions defined on varieties or manifolds. Table of contents showTocToggle("show","hide") 1 Definition and first consequences 2 Examples 2.1 Commutative 2.2 Non-commutative 3 Some facts and defintions 3.3 Commutative 3.4 General Definition and first consequences A ring R is local if it has one (and therefore all) ...

London Underground - ... each day (927 million journeys made 1999-2000). Lines on the Underground can be classified into two types: sub-surface and deep level. The sub-surface lines were dug by the cut-and-cover method, with the tracks running about 5 metres below the surface. Trains on the sub-surface lines have the same loading gauge as British mainline trains. The deep-level or "tube" lines, bored using a tunnelling shield, run about 20 metres below the surface (although this varies considerably ... of opening and the type of tunnelling used. Line Name Map colour Year of opening Type Notes Bakerloo Line Brown 1906 Deep level Central Line Red 1900 Deep level Circle Line Yellow 1884 Sub-surface 1 District Line Green 1868 Sub- ...

Adjoint functors - ... sections considering some of these. Table of contents showTocToggle("show","hide") 1 Motivation 1.1 Ubiquity of adjoint functors 1.2 Deep problems formulated with adjoint functors 1.3 Adjoint functors as solving optimization problems 1.4 The case of partial orders ... Saunders MacLane, any idea such as adjoint functors that occurs widely enough in mathematics should be studied for its own sake. Deep problems formulated with adjoint functors By itself, the generality of the adjoint functor concept isn't a recommendation to most ... homomorphisms R -> S, with S a ring having a multiplicative identity. The morphisms in C must fill in triangles that are commutative diagrams, and preserve multiplicative identity. The assertion is that C has an initial object R -> R*, and R* is ...

Death of a Commuter... - Death of a Commuter Five men occupied their usual places in a first-class carriage, but the sixth place was empty... It is most unusual for the sixth man, Mr. Parador, to be late . The five commuters are wondering what happened to him, when a strange looking man enters the compartment, dressed in black and wearing dark glasses. When he is told that the sixth seat is taken, he replies in a deep sepulchral voice, he won`t be coming. He is right. Parador does not come, and his companions never see ...

Algebraic geometry - ... See live article   Algebraic geometry Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations. When there ... ideals is a central theme of algebraic geometry. Coordinate ring of a variety To every variety V one can associate a commutative ring, the coordinate ring, consisting of all polynomial functions defined on the variety. The prime ideals in this ring correspond ... the subject In the modern view, the correspondence between variety and coordinate ring is turned around: one starts with an abstract commutative ring and defines a corresponding variety via its prime ideals. The prime ideals are first turned into a topological ...