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...

In clustering (also known as unsupervised learning and class discovery), the classes are unknown a priori and need to be identified from the unsupervised data. The cluster analysis is concerned about estimating the number of classes and assigning each observation to a certain class. In this article we discuss a method for clustering via the Laplacian matrix. Also, based on a similar argument, w...

‎let $g$ be a graph without an isolated vertex‎, ‎the normalized laplacian matrix $tilde{mathcal{l}}(g)$‎‎is defined as $tilde{mathcal{l}}(g)=mathcal{d}^{-frac{1}{2}}mathcal{l}(g) mathcal{d}^{-frac{1}{2}}$‎, where ‎$‎mathcal{‎d}‎$ ‎is a‎ diagonal matrix whose entries are degree of ‎vertices ‎‎of ‎$‎g‎$‎‎. ‎the eigenvalues of‎‎$tilde{mathcal{l}}(g)$ are ‎called ‎ ‎ as ‎the ‎normalized laplacian ...

In this paper we define vertical and horizontal Laplace type operators for functions on the total space of a complex Finsler bundle (E, L). We also define the ′′ v-Laplacian for (p, q, r, s)-forms with compact support on E and we get the local expression of this Laplacian explicitly in terms of vertical covariant derivatives with respect to the Chern-Finsler linear connection of (E, L).

The paper introduces complex-valued Laplacians for graphs whose edges are attributed with complex weights and studies the leader-follower formation problem based on complex Laplacians. The main goal is to control the shape of a planar formation of point agents in the plane using simple and linear interaction rules related to complex Laplacians. We present a characterization of complex Laplacian...

A striking result of Bouc gives the decomposition of the representation of the symmetric group on the homology of the matching complex into irreducibles that are self-conjugate. We show how the combinatorial Laplacian can be used to give an elegant proof of this result. We also show that the spectrum of the Laplacian is integral.