Lozenge tilings with free boundaries

We study lozenge tilings of a domain with partially free boundary. In particular, we consider a trapezoidal domain (half hexagon), s.t. the horizontal lozenges on the long side can intersect it anywhere to protrude halfway across. We show that the positions of the horizontal lozenges near the opposite flat vertical boundary have the same joint distribution as the eigenvalues from a Gaussian Unitary Ensemble (the GUE-corners/minors process). We also prove the existence of a limit shape of the height function, which is also a vertically symmetric plane partition. Both behaviors are shown to coincide with those of the corresponding doubled fixed-boundary hexagonal domain. We also consider domains where the different sides converge to $\infty$ at different rates and recover again the GUE-corners process near the boundary.Comment: 27 pages, 4 figures; version 2-- typos fixed, improved proofs and computations, incorporated referee's comments. To appear in Letters in Mathematical Physic

Tableaux and plane partitions of truncated shapes

We consider a new kind of straight and shifted plane partitions/Young tableaux --- ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function.Comment: Accepted to Advances in Applied Mathematics. Final versio

An artificial neural network approach for assigning rating judgements to Italian Small Firms

Based on new regulations of Basel II Accord in 2004, banks and financial nstitutions have now the possibility to develop internal rating systems with the aim of correctly udging financial health status of firms. This study analyses the situation of Italian small firms that are difficult to judge because their economic and financial data are often not available. The intend of this work is to propose a simulation framework to give a rating judgements to firms presenting poor financial information. The model assigns a rating judgement that is a simulated counterpart of that done by Bureau van Dijk-K Finance (BvD). Assigning rating score to small firms with problem of poor availability of financial data is really problematic. Nevertheless, in Italy the majority of firms are small and there is not a law that requires to firms to deposit balance-sheet in a detailed form. For this reason the model proposed in this work is a three-layer framework that allows us to assign ating judgements to small enterprises using simple balance-sheet data.rating judgements, artificial neural networks, feature selection

On the complexity of computing Kronecker coefficients

We study the complexity of computing Kronecker coefficients $g(\lambda,\mu,\nu)$. We give explicit bounds in terms of the number of parts $\ell$ in the partitions, their largest part size $N$ and the smallest second part $M$ of the three partitions. When $M = O(1)$, i.e. one of the partitions is hook-like, the bounds are linear in $\log N$, but depend exponentially on $\ell$. Moreover, similar bounds hold even when $M=e^{O(\ell)}$. By a separate argument, we show that the positivity of Kronecker coefficients can be decided in $O(\log N)$ time for a bounded number $\ell$ of parts and without restriction on $M$. Related problems of computing Kronecker coefficients when one partition is a hook, and computing characters of $S_n$ are also considered.Comment: v3: incorporated referee's comments; accepted to Computational Complexit

Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory

We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems concerning characters of infinite-dimensional unitary group and their $q$-deformations. We study the behavior of uniformly random lozenge tilings of large polygonal domains and find the GUE-eigenvalues distribution in the limit. We also investigate similar behavior for alternating sign matrices (equivalently, six-vertex model with domain wall boundary conditions). Finally, we compute the asymptotic expansion of certain observables in $O(n=1)$ dense loop model.Comment: Published at http://dx.doi.org/10.1214/14-AOP955 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Implicit equations involving the pp-Laplace operator

In this work we study the existence of solutions $u \in W^{1,p}_0(\Omega)$ to the implicit elliptic problem $f(x, u, \nabla u, \Delta_p u)= 0$ in $\Omega$, where $\Omega$ is a bounded domain in $\mathbb R^N$, $N \ge 2$, with smooth boundary $\partial \Omega$, $1< p< +\infty$, and $f\colon \Omega \times \mathbb R \times \mathbb R^N \times \R \to \R$. We choose the particular case when the function $f$ can be expressed in the form $f(x, z, w, y)= \varphi(x, z, w)- \psi(y)$, where the function $\psi$ depends only on the $p$-Laplacian $\Delta_p u$. We also present some applications of our results.Comment: 15 pages; comments are welcom