### A note on linear discrepancy.

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The notion of a transfer (or T -transform) is central in the theory of majorization. For instance, it lies behind the characterization of majorization in terms of doubly stochastic matrices. We introduce a new type of majorization transfer called L-transforms and prove some of its properties. Moreover, we discuss how L-transforms give a new perspective on Ryser’s algorithm for constructing (0; 1)-matrices with given row and column sums.

The Bruhat order is defined in terms of an interchange operation on the set of permutation matrices of order $n$ which corresponds to the transposition of a pair of elements in a permutation. We introduce an extension of this partial order, which we call the stochastic Bruhat order, for the larger class ${\Omega}_{n}$ of doubly stochastic matrices (convex hull of $n\times n$ permutation matrices). An alternative description of this partial order is given. We define a class of special faces of ${\Omega}_{n}$ induced by permutation matrices,...

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