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

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

Large deviations of the free energy in the p-spin glass spherical model

We investigate the behavior of the rare fluctuations of the free energy in the p-spin spherical model, evaluating the corresponding rate function via the G\ue4rtner-Ellis theorem. This approach requires the knowledge of the analytic continuation of the disorder-averaged replicated partition function to arbitrary real number of replicas. In zero external magnetic field, we show via a one-step replica symmetry breaking calculation that the rate function is infinite for fluctuations of the free energy above its typical value, corresponding to an anomalous, superextensive suppression of rare fluctuations. We extend this calculation to nonzero magnetic field, showing that in this case this very large deviation disappears and we try to motivate this finding in light of a geometrical interpretation of the scaled cumulant generating function

Combined effects of electromagnetic fields on immune and nervous responses.

In technologically developed countries, there is concern about hazards from electromagnetic fields (EMFs). Several studies have reported that immune and neuroendocrine systems exert an integrated response to EMF exposure. The aim of this review is to summarize the results of studies on the effect of low and high frequency EMF on immune and neuroendocrine systems on which our research group has been working for several years

Asthma worsened by benzoate contained in some antiasthmatic drugs.

Here, we report our experience on benzoate hypersensitivity. Drug and food additives are known to induce pseudo-allergic reactions such as urticaria, eczema, asthma and rhinitis. These reactions are often under-diagnosed, above all in allergic patients treated with additive containing drugs. On the contrary, attention to additives present in some drug formulations and foods may often permit more correct diagnosis

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

Pathogenesis and clinical approaches to anticonvulsant hypersensitivity syndrome: current state of knowledge.

Anticonvulsant hypersensitivity syndrome (AHS) is a rare, but severe and potentially fatal, adverse reaction that occurs in patients who are treated with commonly used older anticonvulsant drugs (phenytoin, carbamazepine and phenobarbital) and/or with some newer agents (lamotrigine). Paediatric patients are at an increased risk for the development of AHS for the higher incidence of seizure disorder in the first decade of life. Hypersensitivity reactions range from simple maculopapular skin eruptions to a severe life-threatening disorder. AHS is typically associated with the development of skin rash, fever and internal organ dysfunctions. Recent evidence suggests that AHS is the result of a chemotoxic and immunologically-mediated injury, characterized by skin and mucosal bioactivation of antiepileptic drugs and by major histocompatibility complex-dependent clonal expansion of T cells. Early recognition of AHS and withdrawal of anticonvulsant therapy are essential for a successful outcome. In vivo and vitro tests can be helpful for the diagnosis that actually depends essentially on clinical recognition