In this paper, we develop a gradient recovery based linear (GRBL) finite element method (FEM) and Hessian FEM for second-order elliptic equations in non-divergence form. The equation is casted into symmetric weak formulation, which derivatives of the unknown function are involved. We use operators to calculate approximations. Although, thanks low degrees freedom elements, implementation propose...

In contrast to the usual Lipschitz seminorms associated to ordinary metrics on compact spaces, we show by examples that Lipschitz seminorms on possibly non-commutative compact spaces are usually not determined by the restriction of the metric they define on the state space, to the extreme points of the state space. We characterize the Lipschitz norms which are determined by their metric on the ...

The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities:\begin{equation*}\left\{\begin{array}{ll} ([u]_{s,p}^p)^{\sigma-1}(-\Delta)^s_p u = \frac{\lambda}{u^{\gamma}}+u^{ p_s^{*}-1 }\quad \text{in }\Omega,\\ u>0,\;\;\;\;\quad u=0,\;\;\;\;\quad }\mathbb{R}^{N}\setminus \Omega,\end{array} \right. \end{equation*} where $\Omega$ bounded domain ...

We introduce an approximate description of N-qubit state, which contains sufficient information to estimate the expectation value any observable a precision that is upper bounded by ratio suitably-defined seminorm square root number system's identical preparations xmlns:mml="http://www.w3.org/1998/Math/MathML">...

Given a passive tracer distribution $f(x,y)$ , what is the simplest unstirred pattern that may be reached under incompressible advection? This question partially motivated by recent studies of three-dimensional (3-D) magnetic reconnection, in which patterns topological invariant called field line helicity greatly simplify until reaching relaxed state. We test two approaches: variational method ...

Let $\mathbb{B}(\mathcal{H})$ denote the $C^{\ast}$-algebra of all bounded linear operators on a Hilbert space $\big(\mathcal{H}, \langle\cdot, \cdot\rangle\big)$. Given positive operator $A\in\B(\h)$, and number $\lambda\in [0,1]$, seminorm ${\|\cdot\|}_{(A,\lambda)}$ is defined set $\B_{A^{1/2}}(\h)$ in $\B(\h)$ having an $A^{1/2}$-adjoint. The combination sesquilinear form ${\langle \cdot, \...

and Applied Analysis 3 Baeumer et al. 8, 13 have proved existence and uniqueness of a strong solution for 1.2 using the semigroup theory when f x, t, u is globally Lipschitz continuous. Furthermore, when f x, t, u is locally Lipschitz continuous, existence of a unique strong solution has also been shown by introducing the cut-off function. Finite difference methods have been studied in 14–16 fo...

Let A be a positive (semidefinite) bounded linear operator acting on complex Hilbert space \(\big ({\mathcal {H}}, \langle \cdot , \rangle \big )\). The semi-inner product \({\langle x, y\rangle }_A := Ax, \), \(x, y\in {\mathcal {H}}\) induces seminorm \({\Vert \Vert }_A\) \({\mathcal {H}}\). T an A-bounded {H}}\), the A-numerical radius of is given by $$\begin{aligned} \omega _A(T) = \sup \Bi...

Given a discrete sample of event locations, we wish to produce a probability density that models the relative probability of events occurring in a spatial domain. Standard density estimation techniques do not incorporate priors informed by spatial data. Such methods can result in assigning significant positive probability to locations where events cannot realistically occur. In particular, when...

Golomb and Weinberger [1] described a variational approach to interpolation which reduced the problem to minimizing a norm in a reproducing kernel Hilbert space generated by means of a small number of data points. Later, Duchon [2] defined radial basis function interpolants as functions which minimize a suitable seminorm given by a weight in spaces of distributions closely related to Sobolev sp...