Integrable discretizations of a two-dimensional Hamiltonian system with a quartic potential

In this paper, we propose integrable discretizations of a two-dimensional Hamiltonian system with quartic potentials. Using either the method of separation of variables or the method based on bilinear forms, we construct the corresponding integrable mappings for the first three among four integrable cases

Discontinuous Galerkin approximation of linear parabolic problems with dynamic boundary conditions

In this paper we propose and analyze a Discontinuous Galerkin method for a linear parabolic problem with dynamic boundary conditions. We present the formulation and prove stability and optimal a priori error estimates for the fully discrete scheme. More precisely, using polynomials of degree $p\geq 1$ on meshes with granularity $h$ along with a backward Euler time-stepping scheme with time-step $\Delta t$, we prove that the fully-discrete solution is bounded by the data and it converges, in a suitable (mesh-dependent) energy norm, to the exact solution with optimal order $h^p + \Delta t$. The sharpness of the theoretical estimates are verified through several numerical experiments

Image sets of fractional Brownian sheets

Let $B^H = \{ B^H(t), t\in\mathbb{R}^N \}$ be an $(N,d)$-fractional Brownian sheet with Hurst index $H=(H_1,\dotsc,H_N)\in (0,1)^N$. The main objective of the present paper is to study the Hausdorff dimension of the image sets $B^H(F+t)$, $F\subset\mathbb{R}^N$ and $t\in\mathbb{R}^N$, in the dimension case $d<\tfrac{1}{H_1}+\cdots+\tfrac{1}{H_N}$. Following the seminal work of Kaufman (1989), we establish uniform dimensional properties on $B^H$, answering questions raised by Khoshnevisan et al (2006) and Wu and Xiao (2009). For the purpose of this work, we introduce a refinement of the sectorial local-nondeterminism property which can be of independent interest to the study of other fine properties of fractional Brownian sheets.Comment: 14 pages, 1 figur

Is Simple Better? Revisiting Non-linear Matrix Factorization for Learning Incomplete Ratings

Matrix factorization techniques have been widely used as a method for collaborative filtering for recommender systems. In recent times, different variants of deep learning algorithms have been explored in this setting to improve the task of making a personalized recommendation with user-item interaction data. The idea that the mapping between the latent user or item factors and the original features is highly nonlinear suggest that classical matrix factorization techniques are no longer sufficient. In this paper, we propose a multilayer nonlinear semi-nonnegative matrix factorization method, with the motivation that user-item interactions can be modeled more accurately using a linear combination of non-linear item features. Firstly, we learn latent factors for representations of users and items from the designed multilayer nonlinear Semi-NMF approach using explicit ratings. Secondly, the architecture built is compared with deep-learning algorithms like Restricted Boltzmann Machine and state-of-the-art Deep Matrix factorization techniques. By using both supervised rate prediction task and unsupervised clustering in latent item space, we demonstrate that our proposed approach achieves better generalization ability in prediction as well as comparable representation ability as deep matrix factorization in the clustering task.Comment: version

Wigner crystal of a two-dimensional electron gas with a strong spin-orbit interaction

The Wigner-crystal phase of two-dimensional electrons interacting via the Coulomb repulsion and subject to a strong Rashba spin-orbit coupling is investigated. For low enough electronic densities the spin-orbit band splitting can be larger than the zero-point energy of the lattice vibrations. Then the degeneracy of the lower subband results in a spontaneous symmetry breaking of the vibrational ground state. The $60^{\circ}-$rotational symmetry of the triangular (spin-orbit coupling free) structure is lost, and the unit cell of the new lattice contains two electrons. Breaking the rotational symmetry also leads to a (slight) squeezing of the underlying triangular lattice.Comment: 5 pages + appendix, 3 figures, minor improvements to the tex