Fourier Wavefront Reconstruction Over a Rectangular Pupil The iterative and boundary methods developed here for circular and elliptical pupils are based on a method to reconstruct the wavefront over the rectangular pupil. To fully understand this development, it is necessary to first summarize the approach used by Freischlad and Koliopoulos to derive an inverse spatial filter to reconstruct a w...

Let G be a graph on n vertices. Its Laplacian matrix is the n-by-n matrix L(G) = D(G) A(G), where A(G) is the familiar (0, 1) adjacency matrix, and D(G) is the diagonal matrix of vertex degrees. This is primarily an expository article surveying some of the many results known for Laplacian matrices. Its six sections are: Introduction, The Spectrum, The Algebraic Connectivity, Congruence and Equi...

Combinatorial covers of graphs were defined by Chung and Yau. Their main feature is that the spectra of the Combinatorial Laplacian of the base and the total space are related. We extend their definition to directed graphs. As an application, we compute the spectrum of the Combinatorial Laplacian of the homesick random walk RWμ on the line. Using this calculation, we show that the heat kernel o...

The spectrum of a graph has been widely used in graph theory to characterise the properties of a graph and extract information from its structure. It has also been employed as a graph representation for pattern matching since it is invariant to the labelling of the graph. There are however a number of potential drawbacks in using the spectrum as a representation of a graph; Firstly, more than o...

This paper introduces the Laplacian spectral distances, as a function that resembles the usual distance map, but exhibits properties (e.g., smoothness, locality, invariance to shape transformations) that make them useful to processing and analyzing geometric data. Spectral distances are easily defined through a filtering of the Laplacian eigenpairs and reduce to the heat diffusion, wave, bi-har...

We consider the Laplacian in a curved two-dimensional strip of constant width squeezed between two curves, subject to Dirichlet boundary conditions on one of the curves and variable Robin boundary conditions on the other. We prove that, for certain types of Robin boundary conditions, the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Laplacian in a D...

The family G of connected graphs with second largest Laplacian eigenvalue at most θ, where θ = 3.2470 is the largest root of the equation μ−5μ+6μ−1 = 0, is characterized by Wu, Yu and Shu [Y.R. Wu, G.L. Yu and J.L. Shu, Graphs with small second largest Laplacian eigenvalue, European J. Combin. 36 (2014) 190–197]. Let G(a, b, c, d) be a graph with order n = 2a + b + 2c + 3d + 1 that consists of ...

Recently, a method based on Laplacian eigenfunctions was proposed to automatically construct a basis for value function approximation in MDPs. We show that its success may be explained by drawing a connection between the spectrum of the Laplacian and the value function of the MDP. This explanation helps us to identify more precisely the conditions that this method requires to achieve good perfo...

Penrose–Lifshits mushrooms are planar domains coming in nonisometric pairs with the same geodesic length spectrum. Recently S. Zelditch raised the question whether such billiards also have the same eigenvalue spectrum for the Dirichlet Laplacian (conjecturing “no”). Here we show that generically (in the class of smooth domains) the two members of a mushroom pair have different spectra.