Networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors. Social networks, representing people who communicate with each other, are one example. Communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks. Spectral clustering is a popular and computationally feasibl...

For linear operators L , T and nonlinear maps P we describe classes of simple F = I ? between Banach Hilbert spaces, for which no point has more than two preimages. The encompass known examples (homeomorphisms, global folds) the weaker, geometric, hypotheses suggest new ones. operator may be Laplacian with various boundary conditions, as in original Ambrosetti-Prodi theorem, or associated quant...

Spectral decorrelation (transformations) methods have long been used in remote sensing. Transformation of the image data onto eigenvectors that comprise physically meaningful spectral properties (signal) can be used to reduce the dimensionality of hyperspectral images as the number of spectrally distinct signal sources composing a given hyperspectral scene is generally much less than the number...

If G is a graph, its Laplacian is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix. The main thrust of the present article is to prove several Laplacian eigenvector “principles” which in certain cases can be used to deduce the effect on the spectrum of contracting, adding or deleting edges and/or of coalescing vertices. One application is the construction of ...

We present a new algorithm that computes eigenvalues and eigenvectors of a Hermitian positive definite matrix while solving a linear system of equations with Conjugate Gradient (CG). Traditionally, all the CG iteration vectors could be saved and recombined through the eigenvectors of the tridiagonal projection matrix, which is equivalent theoretically to unrestarted Lanczos. Our algorithm capit...

When attempting to develop wavelet transforms for graphs and networks, some researchers have used graph Laplacian eigenvalues and eigenvectors in place of the frequencies and complex exponentials in the Fourier theory for regular lattices in the Euclidean domains. This viewpoint, however, has a fundamental flaw: on a general graph, the Laplacian eigenvalues cannot be interpreted as the frequenc...

For the one-dimensional XXX model under the periodic boundary conditions, we discuss two types of eigenvectors, regular eigenvectors which have finite-valued rapidities satisfying the Bethe ansatz equations, and non-regular eigenvectors which are descendants of some regular eigenvectors under the action of the SU(2) spin-lowering operator. It was pointed out by many authors that the non-regular...