A great deal of classical harmonic analysis is concerned with the properties of the Laplacian L on Euclidean space R and on the sphere Sn−1 in R . One of the key questions which inspired many of the experts on the subject was the following. Given a bounded Borel function m on R , we can define a bounded operator m(L) on L(R) using the functional calculus of self-adjoint operators on a Hilbert s...

Studies of gauge dependent quantities are afflicted with Gribov copies, but Laplacian gauge fixing provides one possible solution to this problem. We present results for the lattice quark propagator in both Landau and Laplacian gauges using standard and improved staggered quark actions. The standard Kogut-Susskind action has errors of O(a) while the improved “Asqtad” action has O(a), O(ag) erro...

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...

Recently, various applications have motivated the study of spectral structures (eigenvalues and eigenfunctions) of the so-called Laplace-Beltrami operator of a manifold and their discrete versions. A popular choice for the discrete version is the so-called Gaussian weighted graph Laplacian which can be applied to point cloud data that samples a manifold. Naturally, the question of stability of ...

In this work we consider the nonlinear graph p-Laplacian and the set of eigenvalues and associated eigenvectors of this operator defined by a variational principle. We prove a unifying nodal domain theorem for the graph p-Laplacian for any p ≥ 1. While for p > 1 the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian (p = 2), the behavior changes...

Pushed by the rising interest in high resolution requirements and high-dimensional applications, the diffusion term in the Laplacian equation drives the condition number of the associated discretized operator to undesirable sizes for standard iterative methods to converge rapidly. In addition, for realistic values of the wavenumber k(x) in (1), the Helmholtz operator H becomes indefinite, destr...

Dirac operators are important geometric operators on a manifold. The Dirac operator DA on the four dimensional Euclidean space M = R is the order one differential operator whose square DA ◦ DA is the Euclidean Laplacian − ∑4 i=1 ∂ψ ∂xi . However, this is not possible unless we allow coefficients for this linear operator to be matrix-valued. Let M = R be the four dimensional Euclidean space with...

We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples. Mathematics Subject Classif...

We develop (within a possibly new) framework spectral analysis and operator theory on (almost) general graphs and use it to study spectral properies of the graph-Laplacian and so-called graph-Dirac-operators. That is, we introduce a Hilbert space structure, being in our framework the direct sum of a node-Hilbert-space and a bond-Hilbert-space, a Dirac operator intertwining these components, and...

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...