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The permutation tensor, also called the Levi-Civita tensor or isotropic tensor of rank 3 (Goldstein 1980, p. 172), is a pseudotensor which is antisymmetric under the ...

The symbol O(x), pronounced "big-O of x," is one of the Landau symbols and is used to symbolically express the asymptotic behavior of a given function. In particular, if n is ...

A braid is an intertwining of some number of strings attached to top and bottom "bars" such that each string never "turns back up." In other words, the path of each string in ...

The orthogonal polynomials defined by c_n^((mu))(x) = _2F_0(-n,-x;;-mu^(-1)) (1) = ((-1)^n)/(mu^n)(x-n+1)_n_1F_1(-n;x-n+1;mu), (2) where (x)_n is the Pochhammer symbol ...

There are at least two meanings on the word congruent in mathematics. Two geometric figures are said to be congruent if one can be transformed into the other by an isometry ...

Let E be an elliptic curve defined over the field of rationals Q(sqrt(-d)) having equation y^2=x^3+ax+b with a and b integers. Let P be a point on E with integer coordinates ...

Two quantities are said to be equal if they are, in some well-defined sense, equivalent. Equality of quantities a and b is written a=b. Equal is implemented in the Wolfram ...

An even permutation is a permutation obtainable from an even number of two-element swaps, i.e., a permutation with permutation symbol equal to +1. For initial set {1,2,3,4}, ...

A formal power series, sometimes simply called a "formal series" (Wilf 1994), of a field F is an infinite sequence {a_0,a_1,a_2,...} over F. Equivalently, it is a function ...

The radical of an ideal a in a ring R is the ideal which is the intersection of all prime ideals containing a. Note that any ideal is contained in a maximal ideal, which is ...