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Let f be a real-valued function defined on an interval [a,b] and let x_0 in (a,b). The four one-sided limits D^+f(x_0)=lim sup_(x->x_0+)(f(x)-f(x_0))/(x-x_0), (1) ...

There are at least two definitions of hypercomplex numbers. Clifford algebraists call their higher dimensional numbers hypercomplex, even though they do not share all the ...

A quadratic form involving n real variables x_1, x_2, ..., x_n associated with the n×n matrix A=a_(ij) is given by Q(x_1,x_2,...,x_n)=a_(ij)x_ix_j, (1) where Einstein ...

Tarski's theorem says that the first-order theory of reals with +, *, =, and > allows quantifier elimination. Algorithmic quantifier elimination implies decidability assuming ...

Any n-dimensional Riemannian manifold can be locally embedded into an (n+1)-dimensional manifold with Ricci curvature Tensor R_(ab)=0. A similar version of the theorem for a ...

A complex vector bundle is a vector bundle pi:E->M whose fiber bundle pi^(-1)(x) is a complex vector space. It is not necessarily a complex manifold, even if its base ...

A subset A of a vector space V is said to be convex if lambdax+(1-lambda)y for all vectors x,y in A, and all scalars lambda in [0,1]. Via induction, this can be seen to be ...

If f:E->B is a fiber bundle with B a paracompact topological space, then f satisfies the homotopy lifting property with respect to all topological spaces. In other words, if ...

A function is called locally integrable if, around every point in the domain, there is a neighborhood on which the function is integrable. The space of locally integrable ...

A topology induced by the metric g defined on a metric space X. The open sets are all subsets that can be realized as the unions of open balls B(x_0,r)={x in X|g(x_0,x)<r}, ...