On Integer Images of Max-plus Linear Mappings

Let us extend the pair of operations (max,+) over real numbers to matrices in the same way as in conventional linear algebra. We study integer images of max-plus linear mappings. The question whether Ax (in the max-plus algebra) is an integer vector for at least one x has been studied for some time but polynomial solution methods seem to exist only in special cases. In the terminology of combinatorial matrix theory this question reads: is it possible to add constants to the columns of a given matrix so that all row maxima are integer? This problem has been motivated by attempts to solve a class of job-scheduling problems. We present two polynomially solvable special cases aiming to move closer to a polynomial solution method in the general case

On the max-algebraic core of a nonnegative matrix

The max-algebraic core of a nonnegative matrix is the intersection of column spans of all max-algebraic matrix powers. Here we investigate the action of a matrix on its core. Being closely related to ultimate periodicity of matrix powers, this study leads us to new modifications and geometric characterizations of robust, orbit periodic and weakly stable matrices.Comment: 27 page

Barabanov norms, Lipschitz continuity and monotonicity for the max algebraic joint spectral radius

We present several results describing the interplay between the max algebraic joint spectral radius (JSR) for compact sets of matrices and suitably defined matrix norms. In particular, we extend a classical result for the conventional algebra, showing that the JSR can be described in terms of induced norms of the matrices in the set. We also show that for a set generating an irreducible semigroup (in a cone-theoretic sense), a monotone Barabanov norm always exists. This fact is then used to show that the max algebraic JSR is locally Lipschitz continuous on the space of compact irreducible sets of matrices with respect to the Hausdorff distance. We then prove that the JSR is Hoelder continuous on the space of compact sets of nonnegative matrices. Finally, we prove a strict monotonicity property for the max algebraic JSR that echoes a fact for the classical JSR

Alemtuzumab long-term immunologic effect: Treg suppressor function increases up to 24 months

To analyze changes in T-helper (Th) subsets, T-regulatory (Treg) cell percentages and function, and mRNA levels of immunologically relevant molecules during a 24-month follow-up after alemtuzumab treatment in patients with relapsing-remitting multiple sclerosis (RRMS)

Z-matrix equations in max algebra, nonnegative linear algebra and other semirings

We study the max-algebraic analogue of equations involving Z-matrices and M-matrices, with an outlook to a more general algebraic setting. We show that these equations can be solved using the Frobenius trace down method in a way similar to that in non-negative linear algebra, characterizing the solvability in terms of supports and access relations. We give a description of the solution set as combination of the least solution and the eigenspace of the matrix, and provide a general algebraic setting in which this result holds.Comment: 21 pages, 1 figur

Croatia in the European Union: what can the citizens expect?

This analysis should contribute towards concretization of the debate about accession of Croatia to the European Union. It purpose is to explain, inform and enable a deeper insight into perspectives which are opening up to Croatia as a candidate country for membership in the EU. The analysis however does not undermine the fears and the prejudices of the citizens but it confronts them. The citizen's fears shoud be taken seriously since accession to the EU represents the only realistic option for the future of Croatia. Furthermore, for its neighbours from the former Yugoslavia Croatia already represents an important bridge towards the EU. Its accession is therefore in the interest of the whole region, as potential regional conflicts can face a long term solution only within the EU