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A real normed algebra, also called a composition algebra, is a multiplication * on R^n that respects the length of vectors, i.e., |x*y|=|x|*|y| for x,y in R^n. The only real ...

Rodrigues' rotation formula gives an efficient method for computing the rotation matrix R in SO(3) corresponding to a rotation by an angle theta about a fixed axis specified ...

A usually simple algorithm or identity. The term is frequently applied to specific orders of Newton-Cotes formulas. The designation "rule n" is also given to the nth ...

A short exact sequence of groups A, B, and C is given by two maps alpha:A->B and beta:B->C and is written 0->A->B->C->0. (1) Because it is an exact sequence, alpha is ...

Let U(P,Q) and V(P,Q) be Lucas sequences generated by P and Q, and define D=P^2-4Q. (1) Let n be an odd composite number with (n,D)=1, and n-(D/n)=2^sd with d odd and s>=0, ...

For a graph vertex x of a graph, let Gamma_x and Delta_x denote the subgraphs of Gamma-x induced by the graph vertices adjacent to and nonadjacent to x, respectively. The ...

A second-tensor rank symmetric tensor is defined as a tensor A for which A^(mn)=A^(nm). (1) Any tensor can be written as a sum of symmetric and antisymmetric parts A^(mn) = ...

For a curve with radius vector r(t), the unit tangent vector T^^(t) is defined by T^^(t) = (r^.)/(|r^.|) (1) = (r^.)/(s^.) (2) = (dr)/(ds), (3) where t is a parameterization ...

An nth-rank tensor in m-dimensional space is a mathematical object that has n indices and m^n components and obeys certain transformation rules. Each index of a tensor ranges ...

The contraction of a tensor is obtained by setting unlike indices equal and summing according to the Einstein summation convention. Contraction reduces the tensor rank by 2. ...