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Given an m×n matrix A, the fundamental theorem of linear algebra is a collection of results relating various properties of the four fundamental matrix subspaces of A. In ...

Consider a second-order ordinary differential equation y^('')+P(x)y^'+Q(x)y=0. If P(x) and Q(x) remain finite at x=x_0, then x_0 is called an ordinary point. If either P(x) ...

A fundamental result of de Rham cohomology is that the kth de Rham cohomology vector space of a manifold M is canonically isomorphic to the Alexander-Spanier cohomology ...

A homology class in a singular homology theory is represented by a finite linear combination of geometric subobjects with zero boundary. Such a linear combination is ...

If two curves phi and psi of multiplicities r_i!=0 and s_i!=0 have only ordinary points or ordinary singular points and cusps in common, then every curve which has at least ...

Let (K,L) be a pair consisting of finite, connected CW-complexes where L is a subcomplex of K. Define the associated chain complex C(K,L) group-wise for each p by setting ...

Given a real m×n matrix A, there are four associated vector subspaces which are known colloquially as its fundamental subspaces, namely the column spaces and the null spaces ...

y=x(dy)/(dx)+f((dy)/(dx)) (1) or y=px+f(p), (2) where f is a function of one variable and p=dy/dx. The general solution is y=cx+f(c). (3) The singular solution envelopes are ...

An algebraic curve over a field K is an equation f(X,Y)=0, where f(X,Y) is a polynomial in X and Y with coefficients in K. A nonsingular algebraic curve is an algebraic curve ...

A hole in a mathematical object is a topological structure which prevents the object from being continuously shrunk to a point. When dealing with topological spaces, a ...