In this work, a non-integer order Airy equation involving Liouville differential operator is considered. Proposing an undetermined integral solution to the left fractional Airy differential equation, we utilize some basic fractional calculus tools to clarify the closed form. A similar suggestion to the right FADE, converts it into an equation in the Laplace domain. An illustration t...

We present an algorithm for approximating the LaplaceBeltrami operator from an arbitrary point cloud obtained from a k-dimensional manifold embedded in the ddimensional space. We show that this PCD Laplace (PointCloud Data Laplace) operator converges to the LaplaceBeltrami operator on the underlying manifold as the point cloud becomes denser. Unlike the previous work, we do not assume that the ...

We define a discrete Laplace-Beltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called “cotan formula”) except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new ...

The study of biharmonic functions under the ordinary (Euclidean) Laplace operator on the open unit disk D in C arises in connection with plate theory, and in particular, with the biharmonic Green functions which measure, subject to various boundary conditions, the deflection at one point due to a load placed at another point. A homogeneous tree T is widely considered as a discrete analogue of t...

Abstract. In this paper we study the almost everywhere convergence of the expansions related to the self-adjoint extension of the LaplaceBeltrami operator on the unit sphere. The sufficient conditions for summability is obtained. The more general properties and representation by the eigenfunctions of the Laplace-Beltrami operator of the Liouville space L 1 is used. For the orders of Riesz means...

einstein, möbius, and proper velocity gyrogroups are relativistic gyrogroups that appear as three different realizations of the proper lorentz group in the real minkowski space-time $bbb{r}^n$. using the gyrolanguage we study their gyroharmonic analysis. although there is an algebraic gyroisomorphism between the three models we show that there are some differences between them. our study focus ...

The d-dimensional discrete Schrödinger operator whose potential is supported on the subspace Zd2 of Zd is considered: H = Ha+VM , where Ha = H0+Va, H0 is the d-dimensional discrete Laplacian, Va is a constant “surface” potential, Va(x) = aδ(x1), x = (x1, x2), x1 ∈ Zd1 , x2 ∈ Zd2 , d1 + d2 = d, and VM (x) = gδ(x1) tanπ(α · x2 + ω) with α ∈ Rd2 , ω ∈ [0, 1). It is proved that if the components of...

We develop new multigrid methods for a class of saddle point problems that include the Stokes system in fluid flow and the Lamé system in linear elasticity as special cases. The new smoothers in the multigrid methods involve optimal preconditioners for the discrete Laplace operator. We prove uniform convergence of the W -cycle algorithm in the energy norm and present numerical results for W -cy...

5 Analysis on weighted graphs 23 5.1 Measures on graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Discrete Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.3 Green’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.4 Integration versus Summation . . . . . . . . . . . . . . . . . . . . . . ...

In this paper, we investigate the possibilities of efficient encoding of UV coordinates associated with vertices of a triangle mesh. Since most parametrization schemes attempt to achieve at least some level of conformality, we exploit the similarity of the shapes of triangles in the mesh and in the parametrization. We propose two approaches building on this idea: first, applying a recently prop...