We address the study of multiparameter quantum groups (MpQGs) at roots unity, namely universal enveloping algebras $U\_\mathbf{q}(\mathfrak{g})$ depending on a matrix parameters $\mathbf{q}=(q\_{ij}){i,j\in I}$. This is performed via construction root vectors and suitable “integral forms” $U\mathbf{q}(\mathfrak{g})$, restricted one—generated by divided powers binomial coefficients—and an unrest...

The PSP model [1], a generalized Surface-Potential (SP) model, has been chosen to be an industry standard for the next generation, 60nm, technology. In order to include quantum effects within surface potential models we use asymptotic analysis techniques [2], [3] applied to the the Density-Gradient equations [4]. Our quantum modified surface potential (SP) results from the ability to obtain a f...

We propose and analyze a numerical method for solving the nonlinear Poisson equation −Δu = f(·, u) on the unit disk with zero Dirichlet boundary conditions. The problem is reformulated as a nonlinear integral equation. We use a Galerkin method with polynomials as approximations. The speed of convergence is shown to be very rapid; and experimentally the maximum error is exponentially decreasing ...

Using a sequential Monte Carlo algorithm, we compute a spectral approximation of the solution of the Poisson equation in dimension 1 and 2. The Feyman-Kac computation of the pointwise solution is achieved using either an integral representation or a modified walk on spheres method. The variances decrease geometrically with the number of steps. A global solution is obtained, accurate up to the i...

A class of α-potentials represented as the sum of modified Green potential and modified Poisson integral are proved to have the growth estimates Rα,l,l x o x β n|x| h |x| −1 at infinity in the upper-half space of the n-dimensional Euclidean space, where the function h |x| is a positive nondecreasing function on the interval 0,∞ satisfying certain conditions. This result generalizes the growth p...

By introducing a new stochastic integral, we investigate the energetics of classical stochastic systems driven by non-Gaussian white noises. In particular, we introduce a decomposition of the total energy difference into the work and the heat for each trajectory, and derive a formula to calculate the heat from experimental data on the dynamics. We apply our formulation and results to a Langevin...

The Laplace–Beltrami operator, a fundamental object associated with Riemannian manifolds, encodes all intrinsic geometry of manifolds and has many desirable properties. Recently, we proposed the point integral method (PIM), a novel numerical method for discretizing the Laplace–Beltrami operator on point clouds (Li et al. in Commun Comput Phys 22(1):228–258, 2017). In this paper, we analyze the ...