Decorous lower bounds for minimum linear arrangement

Minimum Linear Arrangement is a classical basic combinatorial optimization problem from the 1960s, which turns out to be extremely challenging in practice. In particular, for most of its benchmark instances, even the order of magnitude of the optimal solution value is unknown, as testified by the surveys on the problem that contain tables in which the best known solution value often has one more digit than the best known lower bound value. In this paper, we propose a linear-programming based approach to compute lower bounds on the optimum. This allows us, for the first time, to show that the best known solutions are indeed not far from optimal for most of the benchmark instances

Weakly Nonextensive Thermostatistics and the Ising Model with Long--range Interactions

We introduce a nonextensive entropic measure $S_{\chi}$ that grows like $N^{\chi}$, where $N$ is the size of the system under consideration. This kind of nonextensivity arises in a natural way in some $N$-body systems endowed with long-range interactions described by $r^{-\alpha}$ interparticle potentials. The power law (weakly nonextensive) behavior exhibited by $S_{\chi}$ is intermediate between (1) the linear (extensive) regime characterizing the standard Boltzmann-Gibbs entropy and the (2) the exponential law (strongly nonextensive) behavior associated with the Tsallis generalized $q$-entropies. The functional $S_{\chi}$ is parametrized by the real number $\chi \in[1,2]$ in such a way that the standard logarithmic entropy is recovered when $\chi=1$ >. We study the mathematical properties of the new entropy, showing that the basic requirements for a well behaved entropy functional are verified, i.e., $S_{\chi}$ possesses the usual properties of positivity, equiprobability, concavity and irreversibility and verifies Khinchin axioms except the one related to additivity since $S_{\chi}$ is nonextensive. For $1<\chi<2$, the entropy $S_{\chi}$ becomes superadditive in the thermodynamic limit. The present formalism is illustrated by a numerical study of the thermodynamic scaling laws of a ferromagnetic Ising model with long-range interactions.Comment: LaTeX file, 20 pages, 7 figure

The impact of motor symptoms on self-reported anxiety in Parkinson's disease

OBJECTIVE: Anxiety is commonly endorsed in Parkinson's disease (PD) and significantly affects quality of life. The Beck Anxiety Inventory (BAI) is often used but contains items that overlap with common PD motor symptoms (e.g., “hands trembling”). Because of these overlapping items, we hypothesized that PD motor symptoms would significantly affect BAI scores. METHODS: One hundred non-demented individuals with PD and 74 healthy control participants completed the BAI. PD motor symptoms were assessed by the Unified Parkinson's Disease Rating Scale (UPDRS). Factor analysis of the BAI assessed for a PD motor factor, and further analyses assessed how this factor affected BAI scores. RESULTS: BAI scores were significantly higher for PD than NC. A five-item PD motor factor correlated with UPDRS observer-rated motor severity and mediated the PD-control difference on BAI total scores. An interaction occurred, whereby removal of the PD motor factor resulted in a significant reduction in BAI scores for PD relative to NC. The correlation between the BAI and UPDRS significantly declined when controlling for the PD motor factor. CONCLUSIONS: The results indicate that commonly endorsed BAI items may reflect motor symptoms such as tremor instead of, or in addition to, genuine mood symptoms. These findings highlight the importance of considering motor symptoms in the assessment of anxiety in PD and point to the need for selecting anxiety measures that are less subject to contamination by the motor effects of movement disorders.Published versio