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A tautology is a logical statement in which the conclusion is equivalent to the premise. More colloquially, it is formula in propositional calculus which is always true ...

If one looks inside a flat origami without unfolding it, one sees a zigzagged profile, determined by an alternation of "mountain-creases" and "valley-creases." The numbers of ...

A set-like object in which order is ignored, but multiplicity is explicitly significant. Therefore, multisets {1,2,3} and {2,1,3} are equivalent, but {1,1,2,3} and {1,2,3} ...

Suppose for every point x in a manifold M, an inner product <·,·>_x is defined on a tangent space T_xM of M at x. Then the collection of all these inner products is called ...

An ordering of permutations in which each two adjacent permutations differ by the transposition of two elements. For the permutations of {1,2,3} there are two listings which ...

The two recursive sequences U_n = mU_(n-1)+U_(n-2) (1) V_n = mV_(n-1)+V_(n-2) (2) with U_0=0, U_1=1 and V_0=2, V_1=m, can be solved for the individual U_n and V_n. They are ...

Due to Euler's prolific output, there are a great number of theorems that are know by the name "Euler's theorem." A sampling of these are Euler's displacement theorem for ...

Specifying three sides uniquely determines a triangle whose area is given by Heron's formula, K=sqrt(s(s-a)(s-b)(s-c)), (1) where s=1/2(a+b+c) (2) is the semiperimeter of the ...

An interpretation of first-order logic consists of a non-empty domain D and mappings for function and predicate symbols. Every n-place function symbol is mapped to a function ...

The characteristic polynomial is the polynomial left-hand side of the characteristic equation det(A-lambdaI)=0, (1) where A is a square matrix and I is the identity matrix of ...