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Let be a Boolean matrix. The isolation number of is the maximum number of ones in such that no two are in any row or any column (that is they are independent), and no two are in a submatrix of all ones. The isolation number of is a lower bound on the Boolean rank of . A linear operator on the set of Boolean matrices is a mapping which is additive and maps the zero matrix, , to itself. A mapping strongly preserves a set, , if it maps the set into the set and the complement of...
Let be the set of all real matrices. A matrix is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions that preserve or strongly preserve row-dense matrices, i.e., is row-dense whenever is row-dense or is row-dense if and only if is row-dense, respectively. Similarly, a matrix is called a column-dense matrix if every column of is a column-dense vector. At the end, the structure of linear...
We introduce linear transformations of Euclidean topological spaces given by a transformation matrix. Next, we prove selected properties and basic arithmetic operations on these linear transformations. Finally, we show that a linear transformation given by an invertible matrix is a homeomorphism.
We prove a number of theorems concerning various notions used in the theory of continuity of barycentric coordinates.
We consider a commutative ring with identity and a positive integer . We characterize all the 3-tuples of linear transforms over , having the “circular convolution” property, i.eṡuch that for all .
We prove that the linearization functor from the category of Hamiltonian -actions with
group-valued moment maps in the sense of Lu, to the category of ordinary Hamiltonian -
actions, preserves products up to symplectic isomorphism. As an application, we give a
new proof of the Thompson conjecture on singular values of matrix products and extend
this result to the case of real matrices. We give a formula for the Liouville volume of
these spaces and obtain from it a hyperbolic version of the Duflo...
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