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The formula is valid for all index values, and for any n (when n = 0 or n = 1, this is the empty product). However, computing the formula above naively has a time complexity of O( n 2 ) , whereas the sign can be computed from the parity of the permutation from its disjoint cycles in only O( n log( n )) cost.
The trace or tensor contraction, considered as a mapping. The map , representing scalar multiplication as a sum of outer products. The generalized Kronecker delta or multi-index Kronecker delta of order is a type tensor that is completely antisymmetric in its upper indices, and also in its lower indices. Two definitions that differ by a factor ...
The Unicode values of the characters in the tables below, except those shown with pink backgrounds or index values of '–', are obtained by adding the base values from the "U+" header row to the index values in the left column (both values are hexadecimal).
Website. numpy.org. NumPy (pronounced / ˈnʌmpaɪ / NUM-py) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. [3] The predecessor of NumPy, Numeric, was originally created by Jim ...
Tensor contraction. In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor (s) caused by applying the summation convention to a pair of dummy indices that are bound to ...
In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the ...
the Christoffel symbols that describe components of a metric connection; the stack alphabet in the formal definition of a pushdown automaton, or the tape-alphabet in the formal definition of a Turing machine; the Feferman–Schütte ordinal Γ 0; represents: the specific weight of substances; the lower incomplete gamma function
Zernike polynomials have the property of being limited to a range of −1 to +1, i.e. . The radial polynomials are defined as. for an even number of n − m, while it is 0 for an odd number of n − m. A special value is.