Matrices with zero line sums and maximal rank

Matrices of zeros and ones with given line sums and a zero block

We study the existence of (0, 1)-matrices with given line sums and a fixed zero block. An algorithm is given to construct such a matrix which is based on three applications of the well-known Gale–Ryser algorithm for constructing (0, 1)-matrices with given line sums. A characterization in terms of a certain “structure matrix” is proved. Further properties of this structure matrix are also establ...

Modeling Biproportional Apportionment via zero-one matrices with given line sums

In the Biproportional Apportionment Problem (BAP) a matrix of the vote counts of the parties within the constituencies is given, and one has to convert the vote matrix into an integer matrix of seats “as proportional as possible” to it, subject to the following constraints: i) each constituency must be granted its pre-specified number of seats; ii) each party must be allotted the total number o...

Determinants of Certain Classes of Zero-One Matrices with Equal Line Sums

We study the possible determinant values of various classes of n×n zero-one matrices with fixed row and column sums. Some new results, open problems, and conjectures are presented.

Bending Flows for Sums of Rank One Matrices

Non-Maximal Rank Separable States Are A Set Of Measure Zero Within The Set of Non-Maximal Rank States

It is well known that the set of separable pure states is measure 0 in the set of pure states. Herein we extend this fact and show that the set of rank r separable states is measure 0 in the set of rank r states provided r is not maximal rank. Recently quite a few authors have looked at low rank separable and entangled states. (See [1] and the references therein and [2].) Therefore it makes sen...