Plastic number and possible optimal solutions for an Euclidean 2-matching in one dimension

In this work we consider the problem of finding the minimum-weight loop cover of an undirected graph. This combinatorial optimization problem is called 2-matching and can be seen as a relaxation of the traveling salesman problem since one does not have the unique loop condition. We consider this problem both on the complete bipartite and complete graph embedded in a one dimensional interval, the weights being chosen as a convex function of the Euclidean distance between each couple of points. Randomness is introduced throwing independently and uniformly the points in space. We derive the average optimal cost in the limit of large number of points. We prove that the possible solutions are characterized by the presence of "shoelace" loops containing 2 or 3 points of each type in the complete bipartite case, and 3, 4 or 5 points in the complete one. This gives rise to an exponential number of possible solutions scaling as p^N , where p is the plastic constant. This is at variance to what happens in the previously studied one-dimensional models such as the matching and the traveling salesman problem, where for every instance of the disorder there is only one possible solution.Comment: 19 pages, 5 figure

Exact value for the average optimal cost of bipartite traveling-salesman and 2-factor problems in two dimensions

We show that the average cost for the traveling-salesman problem in two dimensions, which is the archetypal problem in combinatorial optimization, in the bipartite case, is simply related to the average cost of the assignment problem with the same Euclidean, increasing, convex weights. In this way we extend a result already known in one dimension where exact solutions are avalaible. The recently determined average cost for the assignment when the cost function is the square of the distance between the points provides therefore an exact prediction $$\overline{E_N} = \frac{1}{\pi}\, \log N$$ for large number of points $2N$. As a byproduct of our analysis also the loop covering problem has the same optimal average cost. We also explain why this result cannot be extended at higher dimensions. We numerically check the exact predictions.Comment: 5 pages, 3 figure

Unified Fock space representation of fractional quantum Hall states

Many bosonic (fermionic) fractional quantum Hall states, such as Laughlin, Moore-Read and Read-Rezayi wavefunctions, belong to a special class of orthogonal polynomials: the Jack polynomials (times a Vandermonde determinant). This fundamental observation allows to point out two different recurrence relations for the coefficients of the permanent (Slater) decomposition of the bosonic (fermionic) states. Here we provide an explicit Fock space representation for these wavefunctions by introducing a two-body squeezing operator which represents them as a Jastrow operator applied to reference states, which are in general simple periodic one dimensional patterns. Remarkably, this operator representation is the same for bosons and fermions, and the different nature of the two recurrence relations is an outcome of particle statistics.Comment: 10 pages, 3 figure

Selberg integrals in 1D random Euclidean optimization problems

We consider a set of Euclidean optimization problems in one dimension, where the cost function associated to the couple of points $x$ and $y$ is the Euclidean distance between them to an arbitrary power $p\ge1$, and the points are chosen at random with flat measure. We derive the exact average cost for the random assignment problem, for any number of points, by using Selberg's integrals. Some variants of these integrals allows to derive also the exact average cost for the bipartite travelling salesman problem.Comment: 9 pages, 2 figure

Average optimal cost for the Euclidean TSP in one dimension

The traveling-salesman problem is one of the most studied combinatorial optimization problems, because of the simplicity in its statement and the difficulty in its solution. We study the traveling salesman problem when the positions of the cities are chosen at random in the unit interval and the cost associated to the travel between two cities is their distance elevated to an arbitrary power $p\in \mathbb{R}$. We characterize the optimal cycle and compute the average optimal cost for every number of cities when the measure used to choose the position of the cities is flat and asymptotically for large number of cities in the other cases. We also show that the optimal cost is a self-averaging quantity, and we test our analytical results with extensive simulations.Comment: 14 pages, 8 figure

Immunopathogenesis of sarcoidosis and risk of malignancy: a lost truth?

The hypothesis of a relationship between sarcoidosis and malignancy was firstly formulated in 1972 by Brincker. He documented an association of sarcoid reactions or sarcoidosis with 19 lymphomas and associated malignancies. Based on various epidemiological studies, for more than 20 years sarcoidosis has been considered as a condition at increased risk for cancer, particularly lymphoproliferative disorders. The existence of a sarcoidosis-lymphoma syndrome was therefore proposed, highlighting, as a potential mechanism, the uncontrolled lymphocyte proliferation and mitotic activity. A reduced ability to eliminate an antigen and chronic inflammation have been suggested as triggering events. Leading to a reduced tumor immune surveillance, a diminished myeloid dendritic cells (mDC) function, despite up-regulated co-stimulatory and maturation markers, was also raised as potential mechanism. However, some subsequent studies have questioned the presence of a close association between the two entities and have explained those previously published as the result of selection bias and misclassification. Recently, a Swedish population-based cohort study documented a significant overall excess incidence of cancer among sarcoidosis patients, especially those with multiple hospitalizations or admission in older age, emphasizing again a potential neoplastic risk. Therefore, currently, whether these patients have an increased risk of developing malignant lesions is still debated. Larger and unbiased studies are needed before drawing definite conclusions

Development and validation of the ID-EC - The ITALIAN version of the identify chronic migraine

Background: Case-finding tools, such as the Identify Chronic Migraine (ID-CM) questionnaire, can improve detection of CM and alleviate its significant societal burden. We aimed to develop and validate the Italian version of the ID-CM (ID-EC) in paper and as a smart app version in a headache clinic-based setting. Methods: The study investigators translated and adapted to the Italian language the original ID-CM questionnaire (ID-EC) and further implemented it as a smart app. The ID-EC was tested in its paper and electronic version in consecutive patients referring to 9 Italian tertiary headache centers for their first in-person visit. The scoring algorithm of the ID-EC paper version was applied by the study investigators (case-finding) and by patients (self-diagnosis), while the smart app provided to patients automatically the diagnosis. Diagnostic accuracy of the ID-EC was assessed by matching the questionnaire results with the interview-based diagnoses performed by the headache specialists during the visit according to the criteria of International Classification of Headache Disorders, III edition, beta version. Results: We enrolled 531 patients in the test of the paper version of ID-EC and 427 in the validation study of the smart app. According to the clinical diagnosis 209 patients had CM in the paper version study and 202 had CM in the smart app study. 79.5% of patients returned valid paper questionnaires, while 100% of patients returned valid and complete smart app questionnaires. The paper questionnaire had a 81.5% sensitivity and a 81.1% specificity for case-finding and a 30.7% sensitivity and 90.7% specificity for self-diagnosis, while the smart app had a 64.9% sensitivity and 90.2% specificity. Conclusions: Our data suggest that the ID-EC, developed and validated in tertiary headache centers, is a valid case-finding tool for CM, with sensitivity and specificity values above 80% in paper form, while the ID-EC smart app is more useful to exclude CM diagnosis in case of a negative result. Further studies are warranted to assess the diagnostic accuracy of the ID-EC in general practice and population-based settings