The almost Mathieu operator is the discrete Schrödinger operator Hα,β,θ on `2(Z) defined via (Hα,β,θf)(k) = f(k+1)+f(k−1)+β cos(2παk+θ)f(k). We derive explicit estimates for the eigenvalues at the edge of the spectrum of the finite-dimensional almost Mathieu operator H α,β,θ. We furthermore show that the (properly rescaled) m-th Hermite function φm is an approximate eigenvector of H α,β,θ, and ...

In applications of principal component analysis (PCA) it has often been observed that the eigenvector with the largest eigenvalue has only nonnegative entries when the vectors of the underlying stochastic process have only nonnegative values. This has been used to show that the coordinate vectors in PCA are all located in a cone. We prove that the nonnegativity of the first eigenvector follows ...

It is very natural to associate the accuracy of the eigenvector with this quantity from a geometric perspective. The indicator in the right-hand side of (1.1) is called (the norm of) the orthogonal complement of the projection of xi onto the space spanned by Q and it can be interpreted as the sine of the canonical angle between the Krylov subspace and an eigenvector. For the moment, we will onl...

The spectrum of a square matrix A, denoted by σ(A) is the set of all eigenvalues of A. The spectral radius of A, denoted by ρ(A) is defined as: ρ(A) = max{|λ| : λ ∈ σ(A)} An eigenvalue of A that is larger in absolute value than any other eigenvalue is called the dominant eigenvalue; a corresponding eigenvector is called a dominant eigenvector. The most commonly encountered vector norm (often si...

Numerous centrality measures have been developed to quantify the importances of nodes in time-independent networks, and many of them can be expressed as the leading eigenvector of some matrix. With the increasing availability of network data that changes in time, it is important to extend such eigenvector-based centrality measures to time-dependent networks. In this paper, we introduce a princi...

In this study, an efficient method to immunize modular networks (i.e., networks with community structure) is proposed. The immunization of networks aims at fragmenting networks into small parts with a small number of removed nodes. Its applications include prevention of epidemic spreading, intentional attacks on networks, and conservation of ecosystems. Although preferential immunization of hub...

In a recent article, Zachary Neal (2011) distinguishes between centrality and power in world city networks and proposes two measures of recursive power and centrality. His effort to clarify oversimplistic interpretations of relational measures of power and position in world city networks is appreciated. However, Neal’s effort to innovate methodologically is based on theoretical reasoning that i...

In this class, we introduced the random walk on graphs. The last lecture shows Perron-Frobenius theory to the analysis of primary eigenvectors which is the stationary distribution. In this lecture we will study the second eigenvector. To analyze the properties of the graph, we construct two matrices: one is (unnormalized) graph Laplacian and the other is normalized graph Laplacian. In the first...

A new modeling and classification method of ultrasonic signals based on empirical mode decomposition(EMD) and neural network is put forward in the paper. Firstly, the original ultrasonic flaw signals are decomposed into a finite number of stationary intrinsic mode functions (IMFs) by EMD, and the Fourier transformation of IMF is made. The next step is to find a set of classification values from...

As the performance of hardware is limited, the focus has been to develop objective, optimized and computationally efficient algorithms for a given task. To this extent, fixed-point and approximate algorithms have been developed and successfully applied in many areas of research. In this paper we propose a feature selection method based on fixed-point algorithm and show its application in the fi...