Inequalities on networks have played important roles in the theory of netwoks. We study several famous inequalities on networks such as Wirtinger’s inequality, Hardy’s inequality, Poincaré-Sobolev’s inequality and the strong isoperimetric inequality, etc. These inequalities are closely related to the smallest eigenvalue of weighted discrete Laplacian. We discuss some relations between these ine...

We study discrete Green’s functions and their relationship with discrete Laplace equations. Several methods for deriving Green’s functions are discussed. Green’s functions can be used to deal with diffusion-type problems on graphs, such as chip-firing, load balancing and discrete Markov chains.

Let H be the discrete Schrödinger operator Hu(n) := u(n − 1) + u(n + 1) + v(n)u(n), u(0) = 0 acting on l(Z) where the potential v is real-valued and v(n) → 0 as n → ∞. Let P be the orthogonal projection onto a closed linear subspace L ⊂ l(Z). In a recent paper E.B. Davies defines the second order spectrum Spec2(H,L) of H relative to L as the set of z ∈ C such that the restriction to L of the op...

We study discrete Green’s functions and their relationship with discrete Laplace equations. Several methods for deriving Green’s functions are discussed. Green’s functions can be used to deal with diffusion-type problems on graphs, such as chip-firing, load balancing and discrete Markov chains.

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 ...

I present an example of a discrete Schrödinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than exponentially. This settles a conjecture of Simon (in the negative). The potential is of von Neumann-Wigner type, with careful navigation around a previously iden...

This report mainly illustrates a way to compute the Laplace-Beltrami-Operator on a Riemannian Manifold and gives information to why and where it is used in the Analysis of 3D Shapes. After a brief introduction, an overview over the necessary properties of manifolds for calculating the Laplacian is given. Furthermore the two operators needed for defining the Laplace-Beltrami-Operator the gradien...

SFB393/05-01 January 2005 Abstract The Laplace-Beltrami operator corresponds to the Laplace operator on curved surfaces. In this paper, we consider an eigenvalue problem for the Laplace-Beltrami operator on subdomains of the unit sphere in R. We develop a residual a posteriori error estimator for the eigenpairs and derive a reliable estimate for the eigenvalues. A global parametrization of the ...

we observe that the variable φ enters the expression in the form of the operator ∂2/∂φ2. This operator—1D Laplace operator with respect to the variable φ—is a Hermitian operator in the space of single-valued functions of the angle φ, because single-valuedness is equivalent to the 2π-periodicity, and Laplace operator is Hermitian in the space of periodic functions. Hence, we have ONB of the eige...

We prove the unique solvability of a mixed boundary value problem for the p -Laplace operator by means of variational methods. Using the obtained results, we construct an iterative procedure for solving the Cauchy problem for the p -Laplace operator.