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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) ...

List of mathematical topics - ... mathematicians, should be listed here. (For a much nicer list of mathematicians, see list of mathematicians.) The list is not necessarily complete or up to date - if you see an article that should be here but isn't (or one that shouldn ... theory -- Combinatorial game theory (history) -- Combinatorial logic -- Combinatorial species -- Combinatorial optimization -- Combinatorial topology -- Combinatorics -- Combinatory logic -- Comma-free code -- Common logarithm -- Commutative -- Commutative algebra -- Commutative diagram -- Commutative ring -- Commutativity -- Commutator -- Compact -- Compact space -- Compactification -- Compactness theorem -- Compiler -- Complement -- Complete Fermi-Dirac integral -- ...

Limit (category theory) - ... every morphism f : X -> Y in J, we have F(f) o X = Y. This situation may be depicted as a commutative diagram, more precisely as a commutative triangle with the vertices L, F(X), and F(Y). Indeed the (usually infinite) collection of all these triangles can ... at all. However, if a functor has two limits then there exists a unique isomorphism between the respective limit objects which commutes with the respective cone maps; this isomorphism is given by the unique factorization from one limit to the other. ...

Lie group - ... under matrix multiplication), for instance the group SO(3) of all rotations in 3-dimensional space. See below for a more complete list of examples. Table of contents showTocToggle("show","hide") 1 Types of Lie groups 2 Homomorphisms and isomorphisms 3 The ... their elements. The Lie algebra associated to a Lie group To every Lie group, we can associate a Lie algebra which completely captures the local structure of the group, at least if the Lie group is connected. This is done as follows. A vector field on a Lie group G is said to be left-invariant if it commutes with left translation, which means the following. Define Lg[f](x)= f(gx) for any analytic function f : G -> ...

Formal power series - ... recursively defined sequences; this is known as the method of generating functions and will be illustrated below. We start with a commutative ring R. We want to define the ring of formal power series over R in the variable X, denoted by ... infinite sequences in R. Define addition of two such sequences by and multiplication by (compare convolution). This turns RN into a commutative ring with multiplicative identity (1,0,0,...). We identify the element a of R with the sequence (a,0,0 ... X]] if R is a field, then R[[X]] is a discrete valuation ring. The metric space (R[[X]], d) is complete. The topology on R[[X]] is equal to the product topology on RN where R is equipped with the ...