Tropical linear algebra with the Lukasiewicz T-norm

The max-Lukasiewicz semiring is defined as the unit interval [0,1] equipped with the arithmetics "a+b"=max(a,b) and "ab"=max(0,a+b-1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Lukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-Lukasiewicz semiring.Comment: 27 page

(K,L)-eigenvectors in max-min algebra

Using the concept of (K,L)-eigenvector, we investigate the structure of the max-min eigenspace associated with a given eigenvalue of a matrix in the max-min algebra (also known as fuzzy algebra). In our approach, the max-min eigenspace is split into several regions according to the order relations between the eigenvalue and the components of x. The resulting theory of (K,L)-eigenvectors, being based on the fundamental results of Gondran and Minoux, allows to describe the whole max-min eigenspace explicitly and in more detail.Comment: New title and abstract, several minor correction

Diffusion tensor and restriction spectrum imaging reflect different aspects of neurodegeneration in Parkinson's disease

To meet the need for Parkinson’s disease biomarkers and evidence for amount and distribution of pathological changes, MRI diffusion tensor imaging (DTI) has been explored in a number of previous studies. However, conflicting results warrant further investigations. As tissue microstructure, particularly of the grey matter, is heterogeneous, a more precise diffusion model may benefit tissue characterization. The purpose of this study was to analyze the diffusion-based imaging technique restriction spectrum imaging (RSI) and DTI, and their ability to detect microstructural changes within brain regions associated with motor function in Parkinson’s disease. Diffusion weighted (DW) MR images of a total of 100 individuals, (46 Parkinson’s disease patients and 54 healthy controls) were collected using b-values of 0–4000s/mm2. Output diffusion-based maps were estimated based on the RSI-model combining the full set of DW-images (Cellular Index (CI), Neurite Density (ND)) and DTI-model combining b = 0 and b = 1000 s/mm2 (fractional anisotropy (FA), Axial-, Mean- and Radial diffusivity (AD, MD, RD)). All parametric maps were analyzed in a voxel-wise group analysis, with focus on typical brain regions associated with Parkinson’s disease pathology. CI, ND and DTI diffusivity metrics (AD, MD, RD) demonstrated the ability to differentiate between groups, with strongest performance within the thalamus, prone to pathology in Parkinson’s disease. Our results indicate that RSI may improve the predictive power of diffusion-based MRI, and provide additional information when combined with the standard diffusivity measurements. In the absence of major atrophy, diffusion techniques may reveal microstructural pathology. Our results suggest that protocols for MRI diffusion imaging may be adapted to more sensitive detection of pathology at different sites of the central nervous system