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In combinatorial mathematics, a partial permutation, or sequence without repetition, on a finite set S is a bijection between two specified subsets of S. That is, it is defined by two subsets U and V of equal size, and a one-to-one mapping from U to V. Equivalently, it is a partial function on S that can be extended to a permutation. [1] [2]
Laplace's expansion by minors for computing the determinant along a row, column or diagonal extends to the permanent by ignoring all signs. [9]For every , = =,,,where , is the entry of the ith row and the jth column of B, and , is the permanent of the submatrix obtained by removing the ith row and the jth column of B.
and it allows to polynomial-time reduce the computation of the permanent of an n×n-matrix with a subset of k or k − 1 rows expressible as linear combinations of another (disjoint) subset of k rows to the computation of the permanent of an (n − k)×(n − k)- or (n − k + 1)×(n − k + 1)-matrix correspondingly, hence having introduced a ...
These partials are customarily given names such as hum, prime, minor third (or tierce), fifth (or quint), octave (or nominal), upper octave, etc. The strike note of the bell, which is determined by three partials (the octave, upper fifth, and the upper octave), is generally close to the pitch of the prime in a well-tuned bell."
The computational complexity of the permanent also has some significance in other aspects of complexity theory: it is not known whether NC equals P (informally, whether every polynomially-solvable problem can be solved by a polylogarithmic-time parallel algorithm) and Ketan Mulmuley has suggested an approach to resolving this question that ...
In music, inharmonicity is the degree to which the frequencies of overtones (also known as partials or partial tones) depart from whole multiples of the fundamental frequency (harmonic series). Acoustically, a note perceived to have a single distinct pitch in fact contains a variety of additional overtones.
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