We consider the problem of the presence of short cycles in the graphs of nonzero elements of matrices which have sublinear rank and nonzero entries on the main diagonal, and analyze the connection of these properties to the rigidity of matrices. In particular, we exhibit a family of matrices which shows that sublinear rank does not imply the existence of triangles. This family can also be used ...

In this work, we present a collision attack on 5 out of 8 rounds of the ECHO256 hash function with a complexity of 2 in time and 2 memory. In this work, we further show that the merge inbound phase can still be solved in the case of hash function attacks on ECHO. As correctly observed by Jean et al., the merge inbound phase of previous hash function attacks succeeds only with a probability of 2...

This is a foundational paper in tropical linear algebra, which is linear algebra over the min-plus semiring. We compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the tropical rank, and we show that they differ in general. Connections to polyhedral geometry, particularly to subdivisions of products of simplices, are emphasized.

We introduce a notion of dimension of max-min convex sets, following the approach of tropical convexity. We introduce a max-min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope. We describe the relation between this rank and the notion of strong regularity in max-min algebra, which is traditionally defined in terms of unique solvabi...

We introduce and study tropical eigenpairs of tensors, a generalization of the tropical spectral theory of matrices. We show the existence and uniqueness of an eigenvalue. We associate to a tensor a directed hypergraph and define a new type of cycle on a hypergraph, which we call an H-cycle. The eigenvalue of a tensor turns out to be equal to the minimal normalized weighted length of H-cycles o...

We consider mixed integer linear programs where free integer variables are expressed in terms of nonnegative continuous variables. When this model only has two integer variables, Dey and Louveaux characterized the intersection cuts that have infinite split rank. We show that, for any number of integer variables, the split rank of an intersection cut generated from a rational lattice-free polyto...

We show that for several notions of rank including tensor rank, Waring rank, and generalized rank with respect to a projective variety, the maximum value of rank is at most twice the generic rank. We show that over the real numbers, the maximum value of the real rank is at most twice the smallest typical rank, which is equal to the (complex) generic rank.

In his paper mentioned in the title, which appears in the same issue of this journal, Mehdi Radjabalipour derives the cyclic decomposition of an algebraic linear transformation. A more general structure theory for linear transformations appears in Irving Kaplansky's lovely 1954 book on infinite abelian groups. We present a translation of Kaplansky's results for abelian groups into the terminolo...

An efficient algorithm for the direct solution of a linear system associated with the discretization of boundary integral equations with oscillatory kernels (in two dimensions) is described without having to compute the complete matrix of the linear system. This algorithm is based on the unitary-weight representation, for which a new construction based on adaptive cross approximation is propose...

The articulated arm coordinate measuring machine (AACMM) is a kind of new coordinate measuring device based on non-Cartesian system. The measuring accuracy of the AACMM can be effectively improved by parameter identification. However, some of the structural parameters are coupling (linearly related), so the structural parameters usually cannot be identified correctly. The Jacobian matrix was ob...