In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multi-linear concept. Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra. In differential geometry an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference to coordinates at all. The same is true in general relativity, of tensor fields...

In the foundations of mathematics, Aczel's anti-foundation axiom is an axiom set forth by Peter Aczel (1988), as an alternative to the axiom of foundation in Zermelo–Fraenkel set theory. It states that every accessible pointed directed graph corresponds to a unique set. In particular, according to this axiom, the graph consisting of a single vertex with a loop corresponds to a set that contains only itself as element, i.e. a Quine atom. A set theory obeying this axiom is necessarily a non-well-founded set theory. == Accessible pointed graphs == An accessible pointed graph is a directed graph with...