The fractional Laplacian and the fractional derivative are two different mathematical concepts (Samko et al, 1987). Both are defined through a singular convolution integral, but the former is guaranteed to be the positive definition via the Riesz potential as the standard Laplace operator, while the latter via the Riemann-Liouville integral is not. It is noted that the fractional Laplacian can ...

We analyze one-dimensional discrete and quasi-continuous linear chains of N >> 1 equidistant and identical mass points with periodic boundary conditions and generalized nonlocal interparticle interactions in the harmonic approximation. We introduce elastic potentials which define by Hamilton’s principle discrete “Laplacian operators” (“Laplacian matrices”) which are operator functions (N ×N -ma...

We study boundary value problems for the Laplacian on a domain Ω consisting of the left half of the Sierpinski Gasket (SG), whose boundary is essentially a countable set of points X. For harmonic functions we give an explicit Poisson integral formula to recover the function from its boundary values, and characterize those that correspond to functions of finite energy. We give an explicit Dirich...

Let (H,B) be an abstract Wiener space and let μs be the Gaussian measure on B with variance s. Let ∆ be the Laplacian (not the number operator), that is, a sum of squares of derivatives associated to an orthonormal basis of H. I will show that the heat operator exp(t∆/2) is a contraction operator from L2(B,μs) to L2(B,μs−t), for all t < s. More generally, the heat operator is a contraction from...

Let M be a complete Riemannian manifold and let Ω•(M) denote the space of differential forms on M . Let d : Ω(M) → Ω(M) be the exterior differential operator and let ∆ = dd + dd be the Laplacian. We establish a sufficient condition for the Schrödinger operator H = ∆ + V (x) (where the potential V (x) : Ω(M) → Ω(M) is a zero order differential operator) to be self-adjoint. Our result generalizes...

Edge detection is a very essential part of image processing, as quality and accuracy of detection determines the success of further processing. We have developed a new self learning technique for edge detection using dictionary comprised of eigenfilters constructed using features of the input image. The dictionary based method eliminates the need of pre or post processing of the image and accou...

We revisit the Rellich inequality from viewpoint of isolating contributions radial and spherical derivatives. This naturally leads to a comparison norms Laplacian Laplace–Beltrami operators with standard Laplacian. In case operator, three-dimensional is most subtle here we improve result Evans Lewis by identifying best constant. Our arguments build on certain identities recently established Wad...

In this paper, by introducing a relativistic Schrödinger tempered fractional p-Laplacian operator (–Δ)p,λs,m, based on the (–Δ + m2)s and Laplacian (Δ λ)β/2, we consider model involving logarithmic nonlinearity. We first establish maximum principle boundary estimate, which play very crucial role in later process. Then obtain radial symmetry monotonicity results using direct method of moving pla...

We derive exact form of the piecewise-linear finite element stiffness matrix on general non-uniform meshes for integral fractional Laplacian operator in one dimension, where derivation is accomplished Fourier transformed space. With such an formulation at our disposal, we are able to numerically study some intrinsic properties commonly used (e.g., graded mesh), particular, examine their seamles...

In this paper, we make the numerical analysis of mild solution which is also an entropy parabolic problem involving \(p(x)-\)Laplacian operator with \(L^1-\) data.