The integral fractional Laplacian of order $s \in (0,1)$ is a nonlocal operator. It known that solutions to the Dirichlet problem involving such an operator exhibit algebraic boundary singularity regardless domain regularity. This, in turn, deteriorates global regularity and as result convergence rate numerical solutions. For finite element discretizations, we derive local error estimates $H^s$...

We formulate a universal characterization of the many-particle quantum entanglement in the ground state of a topologically ordered two-dimensional medium with a mass gap. We consider a disk in the plane, with a smooth boundary of length L, large compared to the correlation length. In the ground state, by tracing out all degrees of freedom in the exterior of the disk, we obtain a marginal densit...

in this paper we study properties of symbols such that these belong to class of symbols sitting insidesm ρ,φ that we shall introduce as the following. so for because hypoelliptic pseudodifferential operatorsplays a key role in quantum mechanics we will investigate some properties of m−hypoelliptic pseudodifferential operators for which define base on this class of symbols. also we consider maxi...

We develop various quantitative estimates for the anisotropic Maxwell system in Lipschitz domains, with a focus on how precisely depend character of domain. pay special attention to trace operators and extension over certain Sobolev spaces. Finally, we provide weak formulation interior scattering problem terms exterior Calder\'on operator, explicit bounds solution incident fields

Suppose that Ω is a bounded convex domain in R whose boundary is a C manifold with corners. We prove that the boundary values (Cauchy data) of eigenfunctions of the Laplacian on Ω with various boundary conditions are quantum ergodic if the classical billiard map β on the ball bundle B(∂Ω) is ergodic. Our proof is based on the classical observation that the boundary values of an interior eigenfu...

In this paper, we develop and analyze a preconditioning technique and an iterative solver for the linear systems resulting from the discretization of second order elliptic problems by the symmetric interior penalty discontinuous Galerkin methods. The main ingredient of our approach is a stable decomposition of the piecewise polynomial discontinuous finite element space of arbitrary order into a...

This paper deals with the isogeometric analysis using B-splines of space rods subject to Kirchhoff-Love hypotheses. A multi-patch isogeometric approach for the numerical analysis of the three-dimensional Kirchhoff-Love rod theory is developed. We use Bezier and B-splines interpolations and we show that they are able to attain very good accuracy for rod structures, particularly for developing a ...

Tomographic reflection is a method that may be used to merge the visual outer surface of a patient with a simultaneous ultrasound scan of the patient’s interior. The technique combines a flat-panel monitor with a halfsilvered mirror such that the image on the monitor is reflected precisely at the proper location within the patient. In this way, the ultrasound image is superimposed in real time ...

In this section we will give an (extremely) brief Introduction to the concept of interior point methods • Logarithmic Barrier Method • Method of Centers We have previously seen methods that follow a path On the boundary of the feasible region (Simplex). As the name suggest, interior point methods instead Follow a path through the interior of the feasible region.