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The Heawood four-color graph is the 25-node planar graph illustrated above that tangles the Kempe chains in Kempe's algorithm and thus provides an example of how Kempe's ...

A requirement necessary for a given statement or theorem to hold. Also called a criterion.

Arrow's paradox, also called Arrow's impossibility theorem or the general possibility theorem, states that perfect democratic voting is impossible, not just in practice but ...

The rule (F,F=>G)/G, where => means "implies," which is the sole rule of inference in propositional calculus. This rule states that if each of F and F=>G is either an axiom ...

A type of mathematical result which is considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Finite ...

Any two rectilinear figures with equal area can be dissected into a finite number of pieces to form each other. This is the Wallace-Bolyai-Gerwien theorem. For minimal ...

The important binomial theorem states that sum_(k=0)^n(n; k)r^k=(1+r)^n. (1) Consider sums of powers of binomial coefficients a_n^((r)) = sum_(k=0)^(n)(n; k)^r (2) = ...

A requirement necessary for a given statement or theorem to hold. Also called a condition.

A metric space X is boundedly compact if all closed bounded subsets of X are compact. Every boundedly compact metric space is complete. (This is a generalization of the ...

Given a Hilbert space H, a *-subalgebra A of B(H) is said to be a von Neumann algebra in H provided that A is equal to its bicommutant A^('') (Dixmier 1981). Here, B(H) ...