In this paper we introduce a multivariate generalized autoregressive conditional heteroskedastic (GARCH) class of models with time-varying eigenvalues. The dynamics the eigenvalues is derived for cases underlying Gaussian and Student’s t-distributed innovations based on general theory dynamic score by Creal, Koopman Lucas (2013) Harvey (2013). resulting eigenvalue GARCH – labeled ‘?-GARCH’ diff...

This paper addresses the impact of the structure of the viral propagation network on the viral prevalence. For that purpose, a new epidemic model of computer virus, known as the node-based SLBS model, is proposed. Our analysis shows that the maximum eigenvalue of the underlying network is a key factor determining the viral prevalence. Specifically, the value range of the maximum eigenvalue is p...

We show lower bounds for the smallest non-trivial eigenvalue, and smallest real portion of an eigenvalue, of the Laplacian of a non-reversible Markov chain in terms of an Evolving set quantity. A myriad of Cheeger-like inequalities follow for non-reversible chains, which even in the reversible case sharpen previously known results. The same argument also produces a new Cheeger-like inequality f...

In this paper, a multilevel correction scheme is proposed to solve the Steklov eigenvalue problem by nonconforming finite element methods. With this new scheme, the accuracy of eigenpair approximations can be improved after each correction step which only needs to solve a source problem on finer finite element space and an Steklov eigenvalue problem on the coarsest finite element space. This co...

This paper discusses finite-element highly efficient calculation schemes for solving eigenvalue problem of electric field. Multigrid discretization is extended to the filter approach for eigenvalue problem of electric field. With this scheme one solves an eigenvalue problem on a coarse grid just at the first step, and then always solves a linear algebraic system on finer and finer grids. Theore...

In the past few years transmission eigenvalues have become an important area of research in inverse scattering theory with active research being undertaken in many parts of the world. Transmission eigenvalues appear in the study of scattering by inhomogeneous media and are closely related to non-scattering waves. Such eigenvalues provide information about material properties of the scattering m...

This article considers the backward error of the solution of polynomial eigenvalue problems expressed as Lagrange interpolants. One of the most common strategies to solve polynomial eigenvalue problems is to linearize, which is to say that the polynomial eigenvalue problem is transformed into an equivalent larger linear eigenvalue problem, and solved using any appropriate eigensolver. Much of t...

In this paper, using variational analysis and optimization techniques, we examine some fundamental analytic properties of Z-eigenvalues of a real symmetric tensor with even order. We first establish that the maximum Z-eigenvalue function is a continuous and convex function on the symmetric tensor space, and so, provide formulas of the convex conjugate function and ε-subdifferential of the maxim...

Article history: Received 5 November 2008 Accepted 3 August 2009 Available online 4 September 2009 Submitted by V. Olshevsky AMS classification: 65F15 65H17