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

High-dimensional manifold of solutions in neural networks: insights from statistical physics

In these pedagogic notes I review the statistical mechanics approach to neural networks, focusing on the paradigmatic example of the perceptron architecture with binary an continuous weights, in the classification setting. I will review the Gardner's approach based on replica method and the derivation of the SAT/UNSAT transition in the storage setting. Then, I discuss some recent works that unveiled how the zero training error configurations are geometrically arranged, and how this arrangement changes as the size of the training set increases. I also illustrate how different regions of solution space can be explored analytically and how the landscape in the vicinity of a solution can be characterized. I give evidence how, in binary weight models, algorithmic hardness is a consequence of the disappearance of a clustered region of solutions that extends to very large distances. Finally, I demonstrate how the study of linear mode connectivity between solutions can give insights into the average shape of the solution manifold.Comment: 22 pages, 9 figures, based on a set of lectures done at the "School of the Italian Society of Statistical Physics", IMT, Lucc

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

Current activities at IITRI on high- temperature protective coatings

Heat resistant protective coatings for use in liquid propellant rocket engine

Structural and functional alterations of the cell nucleus in skeletal muscle wasting: the evidence in situ

The histochemical and ultrastructural analysis of the nuclear components involved in RNA transcription and splicing can reveal the occurrence of cellular dysfunctions eventually related to the onset of a pathological phenotype. In recent years, nuclear histochemistry at light and electron microscopy has increasingly been used to investigate the basic mechanisms of skeletal muscle diseases; the in situ study of nuclei of myofibres and satellite cells proved to be crucial for understanding the pathogenesis of skeletal muscle wasting in sarcopenia, myotonic dystrophy and laminopathies

In vitro models of biological barriers for nanomedical research

Nanoconstructs developed for biomedical purposes must overcome diverse biological barriers before reaching the target where playing their therapeutic or diagnostic function. In vivo models are very complex and unsuitable to distinguish the roles plaid by the multiple biological barriers on nanoparticle biodistribution and effect; in addition, they are costly, time-consuming and subject to strict ethical regulation. For these reasons, simplified in vitro models are preferred, at least for the earlier phases of the nanoconstruct development. Many in vitro models have therefore been set up. Each model has its own pros and cons: conventional 2D cell cultures are simple and cost-effective, but the information remains limited to single cells; cell monolayers allow the formation of cell-cell junctions and the assessment of nanoparticle translocation across structured barriers but they lack three-dimensionality; 3D cell culture systems are more appropriate to test in vitro nanoparticle biodistribution but they are static; finally, bioreactors and microfluidic devices can mimicking the physiological flow occurring in vivo thus providing in vitro biological barrier models suitable to reliably assess nanoparticles relocation. In this evolving context, the present review provides an overview of the most representative and performing in vitro models of biological barriers set up for nanomedical research

Identifying pathological biomarkers: histochemistry still ranks high in the omics era

In recent years, omic analyses have been proposed as possible approaches to diagnosis, in particular for tumours, as they should be able to provide quantitative tools to detect and measure abnormalities in gene and protein expression, through the evaluation of transcription and translation products in the abnormal vs normal tissues. Unfortunately, this approach proved to be much less powerful than expected, due to both intrinsic technical limits and the nature itself of the pathological tissues to be investigated, the heterogeneity deriving from polyclonality and tissue phenotype variability between patients being a major limiting factor in the search for unique omic biomarkers. Especially in the last few years, the application of refined techniques for investigating gene expression in situ has greatly increased the diagnostic/prognostic potential of histochemistry, while the progress in light microscopy technology and in the methods for imaging molecules in vivo have provided valuable tools for elucidating the molecular events and the basic mechanisms leading to a pathological condition. Histochemical techniques thus remain irreplaceable in pathologist's armamentarium, and it may be expected that even in the future histochemistry will keep a leading position among the methodological approaches for clinical pathology