In this paper we study in detail some spectral properties of the magnetic discrete Laplacian. We identify its form-domain, characterize the absence of essential spectrum and provide the asymptotic eigenvalue distribution.

We determine the joint distribution of the lengths of, and angles between, the N shortest lattice vectors in a random n-dimensional lattice as n → ∞. Moreover we interpret the result in terms of eigenvalues and eigenfunctions of the Laplacian on flat tori. Finally we discuss the limit distribution of any finite number of successive minima of a random n-dimensional lattice as n → ∞.

Various ensembles of random matrices with independent entries are analyzed by the replica formalism in the large- N limit. A result on the Laplacian random matrix with Wigner-rescaling is generalized to arbitrary probability distribution.

In order to achieve a multiple resolution with multiple directional feature analysis, a Laplacian pyramid is combined with directional filter bank. This requires the analysis on the design of both the filter in the Laplacian pyramid and the overall directional decomposition structure. Our analysis shows that the energy distribution of the filter in the Laplacian pyramid should match with the an...

A new implementation of the surface Laplacian derivation (SLD) method is described which reconstructs a realistically shaped, local scalp surface geometry using measured electrode positions, generates a local spectral-interpolated potential distribution function, and estimates the surface Laplacian values through a local planar parametric space using a stable numerical method combining Taylor e...

In this paper, we investigate the use of heat kernels as a means of embedding graphs in a pattern space. We commence by performing the spectral decomposition on the graph Laplacian. The heat kernel of the graph is found by exponentiating the resulting eigensystem over time. By equating the spectral heat kernel and its Gaussian form we are able to approximate the geodesic distance between nodes ...

We derive the optimal -differentially private mechanism for a general two-dimensional real-valued (histogram-like) query function under a utility-maximization (or cost-minimization) framework for the ` cost function. We show that the optimal noise probability distribution has a correlated multidimensional staircase-shaped probability density function. Compared with the Laplacian mechanism, we s...

Product graphs have been gainfully used in literature to generate mathematical models of complex networks which inherit properties of real networks. Realizing the duplication phenomena imbibed in the definition of corona product of two graphs, we define corona graphs. Given a small simple connected graph which we call basic graph, corona graphs are defined by taking corona product of the basic ...

Let mGI denote the number of Laplacian eigenvalues a graph G in an interval I and let α(G) independence G. In this paper, we determine classes graphs that satisfy condition mG[0,n−α(G)]=α(G) when α(G)=2 α(G)=n−2, where n is order When α(G)=2, G≅K1∇Kn−m∇Km−1 for some m≥2. there are two types B(p,q,r) B′(p,q,r) = p + q r 2, which call binary star graphs. Also, show with determined by their spectra.

Spectral compression of triangle meshes has shown good results in practice, but there has been little or no theoretical support for the optimality of this compression. We show that for certain classes of geometric mesh models, spectral decomposition using the eigenvectors of the symmetric Laplacian of the connectivity graph is equivalent to principal component analysis. The key component of the...