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The direct sum of modules A and B is the module A direct sum B={a direct sum b|a in A,b in B}, (1) where all algebraic operations are defined componentwise. In particular, ...

For elliptic curves over the rationals Q, the group of rational points is always finitely generated (i.e., there always exists a finite set of group generators). This theorem ...

In a monoid or multiplicative group where the operation is a product ·, the multiplicative inverse of any element g is the element g^(-1) such that g·g^(-1)=g^(-1)·g=1, with ...

Consider a power series in a complex variable z g(z)=sum_(n=0)^inftya_nz^n (1) that is convergent within the open disk D:|z|<R. Convergence is limited to within D by the ...

The neighborhood complex N(G) of a locally finite graph G is defined as the abstract simplicial complex formed by the subsets of the neighborhoods of all vertices of G.

Let pi_n(x)=product_(k=0)^n(x-x_k), (1) then f(x)=f_0+sum_(k=1)^npi_(k-1)(x)[x_0,x_1,...,x_k]+R_n, (2) where [x_1,...] is a divided difference, and the remainder is ...

By way of analogy with the eban numbers, oban numbers are defined as numbers whose English names do not contain the letter "o" (i.e., "o" is banned). Note that this ...

Let T be a linear operator on a separable Hilbert space. The spectrum sigma(T) of T is the set of lambda such that (T-lambdaI) is not invertible on all of the Hilbert space, ...

Ore algebra is an algebra of noncommutative polynomials representing linear operators for functional equations such as linear differential or difference equations. Ore ...

A group that coincides with its commutator subgroup. If G is a non-Abelian group, its commutator subgroup is a normal subgroup other than the trivial group. It follows that ...