Abstract:
We analyze the random Euclidean bipartite matching problem on the hypertorus in $d$ dimensions with quadratic cost and we derive the two--point correlation function for the optimal matching, using a proper ansatz introduced by Caracciolo et al. to evaluate the average optimal matching cost. We consider both the grid--Poisson matching problem and the Poisson--Poisson matching problem. We also show that the correlation function is strictly related to the Green's function of the Laplace operator on the hypertorus.

Abstract:
We discuss the equivalence relation between the Euclidean bipartite matching problem on the line and on the circumference and the Brownian bridge process on the same domains. The equivalence allows us to compute the correlation function and the optimal cost of the original combinatoric problem in the thermodynamic limit; moreover, we solve also the minimax problem on the line and on the circumference. The properties of the average cost and correlation functions are discussed.

Abstract:
We propose a new approach for the study of the quadratic stochastic Euclidean bipartite matching problem between two sets of $N$ points each, $N\gg 1$. The points are supposed independently randomly generated on a domain $\Omega\subset\mathbb R^d$ with a given distribution $\rho(\mathbf x)$ on $\Omega$. In particular, we derive a general expression for the correlation function and for the average optimal cost of the optimal matching. A previous ansatz for the matching problem on the flat hypertorus is obtained as particular case.

Abstract:
We propose a unifying picture where the notion of generalized entropy is related to information theory by means of a group-theoretical approach. The group structure comes from the requirement that an entropy be well defined with respect to the composition of independent systems, in the context of a recently proposed generalization of the Shannon-Khinchin axioms. We associate a general information functional to each member of a large class of non-extensive entropies, satisfying the additivity property on a set of independent systems on the basis of the underlying group law. At the same time, we also show that the Einstein likelihood function naturally emerges as a byproduct of our informational interpretation of nonadditive entropies. These results confirm the adequacy of composable entropies both in physical and social science contexts.

Abstract:
We propose a theorem that extends the classical Lie approach to the case of fractional partial differential equations (fPDEs) of the Riemann--Liouville type in (1+1) dimensions.

Abstract:
A general theoretical approach for the determination of Lie symmetries of fractional order differential equations, with an arbitrary number of independent variables, is proposed. We prove a theorem for the existence of vector fields acting as infinitesimal generators of Lie groups of transformations which leave invariant a given equation. As an application of the theory, a symmetry reduction technique for a $(N+1)$-dimensional fractional equation is developed.

Abstract:
We introduce three deformations, called $\alpha$-, $\beta$- and $\gamma$-deformation respectively, of a $N$-body probabilistic model, first proposed by Rodr\'iguez et al. (2008), having $q$-Gaussians as $N\to\infty$ limiting probability distributions. The proposed $\alpha$- and $\beta$-deformations are asymptotically scale-invariant, whereas the $\gamma$-deformation is not. We prove that, for both $\alpha$- and $\beta$-deformations, the resulting deformed triangles still have $q$-Gaussians as limiting distributions, with a value of $q$ independent (dependent) on the deformation parameter in the $\alpha$-case ($\beta$-case). In contrast, the $\gamma$-case, where we have used the celebrated $Q$-numbers and the Gauss binomial coefficients, yields other limiting probability distribution functions, outside the $q$-Gaussian family. These results suggest that scale-invariance might play an important role regarding the robustness of the $q$-Gaussian family.

Abstract:
We propose a simple yet very predictive form, based on a Poisson's equation, for the functional dependence of the cost from the density of points in the Euclidean bipartite matching problem. This leads, for quadratic costs, to the analytic prediction of the large $N$ limit of the average cost in dimension $d=1,2$ and of the subleading correction in higher dimension. A non-trivial scaling exponent, $\gamma_d=\frac{d-2}{d}$, which differs from the monopartite's one, is found for the subleading correction. We argue that the same scaling holds true for a generic cost exponent in dimension $d>2$.

Abstract:
a survey of the research on language acquisition is presented, in which the major trends, controversies and findings of the last 30 years are highlighted. the study of language acquisition carried out in brazil is situated within the broader context of the research in this area. the perspectives of a theory of language acquisition are considered.

Abstract:
A survey of the research on language acquisition is presented, in which the major trends, controversies and findings of the last 30 years are highlighted. The study of language acquisition carried out in Brazil is situated within the broader context of the research in this area. The perspectives of a theory of language acquisition are considered.