R. B. Bapat and T.E.S. Raghavan. Nonnegative Matrices and Applications, volume 64 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1997. Home/Search Document Not in Database Summary Related Articles Check |
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The Disparity between Work and Entropy in Cryptology - Pliam (1999) (Correct)
....4 The Hardy Littlewood P olya theorem originally appeared in 1929 [HLP29] The Birkhoff theorem first appeared in 1946 [Bir46] but was independently discovered in 1953 by von Neumann [von53] who applied it to a problem in game theory. It is sometimes called the Birkhoff von Neumann theorem [BR97]. Recall that the convex hull of a set of vectors B = fv 1 ; v 2 ; v k g is defined by Omega Gamma B) k X i=1 p i v i fi fi fi fi fi k X i=1 p i = 1 ) Thus Birkhoff s theorem identifies the set of doubly stochastic matrices with the convex hull of permutation matrices. ....
....OE y, then x = Dy, where D is a nontrivial convex sum of permutation matrices. However, the converse is not in general true, because for any doubly stochastic D, e = De; but e 6OE e; where e = 1; 1; 1) Strict majorization leads to useful strict inequalities, for example (see [MO79] or [BR97]) x OE y ) kxk 2 kyk 2 : It is natural to seek a kind of converse to proposition 1. This is taken up in [Pli99] 1.3 Work and Majorization It is useful to generalize the definition of work factor and guesswork for arbitrary vectors in R n . Definition 3 For x 2 R n and real 0 ff ....
R. B. Bapat and T.E.S. Raghavan. Nonnegative Matrices and Applications, volume 64 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1997.
The Disparity between Work and Entropy in Cryptology - Pliam (1999) (Correct)
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R. B. Bapat and T.E.S. Raghavan. Nonnegative Matrices and Applications, volume 64 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1997.
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