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In 1926, Van der Waerden conjectured that the minimum permanent among all n × n doubly stochastic matrices is n!/n n, achieved by the matrix for which all entries are equal to 1/n. [18] Proofs of this conjecture were published in 1980 by B. Gyires [ 19 ] and in 1981 by G. P. Egorychev [ 20 ] and D. I. Falikman; [ 21 ] Egorychev's proof is an ...
While the compression operator maps the class of 1-semi-unitary matrices to itself and the classes of unitary and 2-semi-unitary ones, the compression-closure of the 1-semi-unitary class (as well as the class of matrices received from unitary ones through replacing one row by an arbitrary row vector — the permanent of such a matrix is, via ...
The hafnian of a symmetric matrix is defined as = {,},, where is the set of all partitions of the set {,, …,} into subsets of size . [2] [3]This definition is similar to that of the Pfaffian, but differs in that the signatures of the permutations are not taken into account.
Any square matrix = can be viewed as the adjacency matrix of a directed graph, with representing the weight of the edge from vertex to vertex .Then, the permanent of is equal to the sum of the weights of all cycle-covers of the graph; this is a graph-theoretic interpretation of the permanent.
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Defense Secretary Pete Hegseth ordered an immediate pause on gender-affirming medical care procedures for all active-duty service members in a memo that was addressed to senior Pentagon leadership ...