This paper exploits the properties of the commute time to develop a graphspectral method for image segmentation. Our starting point is the lazy random walk on the graph, which is determined by the heat-kernel of the graph and can be computed from the spectrum of the graph Laplacian. We characterise the random walk using the commute time between nodes, and show how this quantity may be computed ...

In this paper, we study the following fractional Schrödinger-poisson systems involving fractional Laplacian operator { (−∆)su+ V (|x|)u+ φ(|x|, u) = f(|x|, u), x ∈ R3, (−∆)tφ = u2, x ∈ R3, (1) where (−∆)s(s ∈ (0, 1)) and (−∆)t(t ∈ (0, 1)) denotes the fractional Laplacian. By variational methods, we obtain the existence of a sequence of radial solutions. c ©2016 All rights reserved.

We study ∗-differential calculi over compact quantum groups in the sense of S.L. Woronowicz. Our principal results are the construction of a Hodge operator commuting with the Laplacian, the derivation of a corresponding Hodge decomposition of the calculus of forms, and, for Woronowicz’ first calculus, the calculation of the eigenvalues of the Laplacian.

Let T be a time scale. We study the existence of positive solutions for the higher-order p-Laplacian dynamic delay differential equations on time scales. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear four-point singular boundary value problem with p-Laplacian operator are obtained. Mathematics Subject Classification: 34B16, 3...

In this short note, we prove the convexity of the first eigenfunction of the drifting Laplacian operator with zero Dirichlet boundary value provided a suitable assumption to the drifting term is added. After giving a gradient estimate, we then use the convexity of the first eigenfunction to get a lower bound of the difference of the first and second eigenvalues of the drifting Laplacian.

Keywords: Dirichlet boundary value problem p-Laplacian Topological degree theory Critical point theory Weak solution a b s t r a c t In this work, by virtue of topological degree theory and critical point theory, we are mainly concerned with the existence of weak solutions for a Dirichlet boundary value problem with the p-Laplacian operator.

We consider Laplacian in a planar strip with Dirichlet boundary condition on the upper boundary and with frequent alternation boundary condition on the lower boundary. The alternation is introduced by the periodic partition of the boundary into small segments on which Dirichlet and Neumann conditions are imposed in turns. We show that under the certain condition the homogenized operator is the ...

Let (M,g) be a complete non-compact Riemannian manifold. We consider operators of the form ∆g + V , where ∆g is the non-negative Laplacian associated with the metric g, and V a locally integrable function. Let ρ : (M̂ , ĝ) → (M,g) be a Riemannian covering, with Laplacian ∆ĝ and potential V̂ = V ◦ ρ. If the operator ∆ + V is non-negative on (M,g), then the operator ∆ĝ + V̂ is non-negative on (M̂, ĝ)...

We discuss the topological properties of a two-dimensional free Abelian gauge theory in the framework of BRST cohomology. We derive the conserved and nilpotent BRST and co-BRST charges and express the Hodge decomposition theorem in terms of these charges and the Laplacian operator. It is because of the topological nature of the free U(1) gauge theory that the Laplacian operator goes to zero whe...

We use the spectra of Dirac type operators on the sphere S to produce sharp L inequalities on the sphere. These operators include the Dirac operator on S, the conformal Laplacian and Paenitz operator. We use the Cayley transform, or stereographic projection, to obtain similar inequalities for powers of the Dirac operator and their inverses in R.