We present an efficient discrete Hodge decomposition for velocity fields defined over irregular domains in two and three dimensions using a virtual node framework. The method is designed for use in the exact projection discretization of incompressible flow. We leverage the Poisson framework initially developed in [1] and [2]. This approach uses a signed distance function to represent the irregular domain embedded in a Cartesian grid and uses a variational approach to create a symmetric positive definite linear system. We present a novel modification to the previous approach that yields a 5-point stencil (7-point in 3D) across the entire computational domain. The original algorithm required a 9-point stencil (27-point in 3D) near the embedded irregular boundary. We show that this new condensed stencil enables a decomposition of the form A = G T M 1 G, where M is a diagonal weighting matrix and G and D = G T are diagonal scalings of the standard central-difference gradient and divergence operators. We use this factored form as the basis of our discrete Hodge decomposition and show that this can be readily used for exact projection in incompressible flow. Numerical experiments suggest our method is second-order in L ¥ for pressures and first-order in L ¥ , second-order in L 1 for velocities.

Electromagnetic scattering by a dissipative chiral obstacle in achiral surrounding leads to a transmission problem. The transmission problem is reduced to a single integral equation over the boundary of the obstacle with one unknown tangential vector field. The equation is shown to be uniquely solvable except for some values of the material parameters and the frequency. The principal symbol of a boundary integral operator is also calculated.

An orderly procedure of constructing families of and schemes, to approximate homogeneous conservation laws, was introduced by S. Osher and S. Chakravarthy Among these schemes, we are interested in the entropy consistence of the semi-discrete schemes in the context of approximating scalar conservation laws. In general schemes, for 0 < 1 2 and m = 2, 3 , 8, are 2m -2 order accurate methods. However when = 2 m 2m m -1 , we obtain one order higher, i.e. 2m -1 order, accurate schemes. Although, for the homogeneous conservation laws, Osher and Chakravarthy were able to show the total variation diminishing (TVD) property of these schemes, the entropy convergence of the schemes has been open. In this paper, for all admissible values of and m = 2, we extend schemes to approximate the non-homogeneous convex conservation laws, which in general are total variation bounded (TVB). Finally, using one of our convergence criteria [22], we have established the entropy convergence for both the original and extended schemes.

We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval. Here the eigenfunctions are nonzero solutions exponentially decaying at infinity. We prove that at any discrete eigenvalue the differential equations are integrable in the setting of differential Galois theory under general assumptions. Our result is illustrated with two examples for a stationary Schroedinger equation having a generalized Hulthen potential and an eigenvalue problem for a traveling front in the Allen-Cahn equation.

In this paper, we prove a version of weighted inequalities of exponential type for fractional integrals with sharp constants in any domain of finite measure in $\mathbb{R}^{n}$. Using this we prove a sharp singular Adams inequality in high order Sobolev spaces in bounded domain at critical case. Then we prove sharp singular Adams inequalities for high order derivatives on unbounded domains. Our results extend the singular Moser-Trudinger inequalities of first order in \cite{Ad2, R, LR, AdY} to the higher order Sobolev spaces $W^{m,\frac{n}{m}}$ and the results of \cite{RS} on Adams type inequalities in unbounded domains to singular case. Our singular Adams inequality on $W^{2,2}(\mathbb{R}^{4})$ with standard Sobolev norm at the critical case settles a unsolved question remained in \cite{Y}.

In this paper, we investigate the wavelet coefficients for function spacesIn particular, Bernstein type inequalities associated with wavelets are established.We obtained an sharp inequality of Bernstein type for splines and a lower bound for the quantity C k,p (ψ) with ψ being the semiorthogonal spline wavelets.We also study the asymptotic behavior of wavelet coefficients for both the family of Daubechies orthonormal wavelets and the family of semiorthogonal spline wavelets.Comparison of these two families is done by using the quantity C k,p (ψ).

In this paper, we study the Lebesgue property for convex risk measures on the class of bounded c`adl`ag processes. For that, we characterize the compact subsets of a family of bounded variation processes, which is, of course, the topological dual of the bounded c`adl`ag processes, in an appropriate topology. We show that the Lebesgue property can be characterized in several equivalent ways.

In this work we consider a general class of Schrdinger type operators, associated to multi-quasi-elliptic symbols introduced by Buzano and Ziggioto in We develop their quasiclassical analysis and we obtain a uniform asymptotic formula for their counting function N( ), in the sense that it holds as + and for all 0 < 1.

We consider the inverse problem of reconstructing an unknown coefficient in a second order hyperbolic equation from partial (on part of the boundary) dynamic boundary measurements. In this paper we prove that the knowledge of the partial Cauchy data for this class of hyperbolic PDE on any open subset of the boundary determines explicitly the coefficient c provided that c is known outside a bounded domain. Then, through construction of appropriate test functions by a geometrical control method, we derive a formula for calculating the coefficient c from the knowledge of the difference between the local Dirichlet to Neumann maps.

In this paper , the author proves a Liouville type theorem for some Hessian entire inequality with sub-lower-critical exponent, via suitable choices of test functions and the argument of integration by parts .