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An infinite sequence of homomorphisms of modules or additive Abelian groups ...->C_(i+1)->^(d_(i+1))C_i->^(d_i)C_(i-1)->... (1) such that, for all indices i in Z, ...

A ring defined on a singleton set {*}. The ring operations (multiplication and addition) are defined in the only possible way, *·*=*, (1) and *+*=*. (2) It follows that this ...

A square matrix A is said to be unipotent if A-I, where I is an identity matrix is a nilpotent matrix (defined by the property that A^n is the zero matrix for some positive ...

The extension ring obtained from a commutative unit ring (other than the trivial ring) when allowing division by all non-zero divisors. The ring of fractions of an integral ...

If, in an interval of x, sum_(r=1)^(n)a_r(x) is uniformly bounded with respect to n and x, and {v_r} is a sequence of positive non-increasing quantities tending to zero, then ...

A dynamical system in which the phase space volume contracts along a trajectory. This means that the generalized divergence is less than zero, (partialf_i)/(partialx_i)<0, ...

If A is an n×n square matrix and lambda is an eigenvalue of A, then the union of the zero vector 0 and the set of all eigenvectors corresponding to eigenvalues lambda is ...

Sequences x_n^((1)), x_n^((2)), ..., x_n^((k)) are linearly dependent if constants c_1, c_2, ..., c_k (not all zero) exist such that sum_(i=1)^kc_ix_n^((i))=0 for n=0, 1, ....

Pathological functions that are continuous but differentiable only on a set of points of measure zero are sometimes known as monsters of real analysis. Examples include the ...

The set of nilpotent elements in a commutative ring is an ideal, and it is called the nilradical. Another equivalent description is that it is the intersection of the prime ...