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If A is a unital Banach algebra where every nonzero element is invertible, then A is the algebra of complex numbers.

In a chain complex of modules ...->C_(i+1)->^(d_(i+1))C_i->^(d_i)C_(i-1)->... the module Z_i of i-cycles is the kernel of d_i, which is a submodule of C_i.

Let K be a finite complex, let h:|K|->|K| be a continuous map. If Lambda(h)!=0, then h has a fixed point.

An algorithm which isolates roots in the complex plane by generalizing one-dimensional bracketing.

A complex manifold for which the exterior derivative of the fundamental form Omega associated with the given Hermitian metric vanishes, so dOmega=0. In other words, it is a ...

A gadget defined for complex vector bundles. The Chern classes of a complex manifold are the Chern classes of its tangent bundle. The ith Chern class is an obstruction to the ...

The field of all rational and irrational numbers is called the real numbers, or simply the "reals," and denoted R. The set of real numbers is also called the continuum, ...

In general, there is no unique matrix solution A to the matrix equation y=Ax. Even in the case of y parallel to x, there are still multiple matrices that perform this ...

The operator partial^_ is defined on a complex manifold, and is called the 'del bar operator.' The exterior derivative d takes a function and yields a one-form. It decomposes ...

Division by zero is the operation of taking the quotient of any number x and 0, i.e., x/0. The uniqueness of division breaks down when dividing by zero, since the product ...