Perron-Frobenius operators and their eigendecompositions are increasingly being used as tools of global analysis for higher dimensional systems. The numerical computation of large, isolated eigenvalues and their corresponding eigenfunctions can reveal important persistent structures such as almostinvariant sets, however, often little can be said rigorously about such calculations. We attempt to...

We present a computational method for solving a class of boundary-value problems in Sturm-Liouville form. The algorithms are based on global polynomial collocation methods and produce discrete representations of the eigenfunctions. Error control is performed by evaluating the eigenvalue problem residuals generated when the eigenfunctions are interpolated to a finer discretization grid; eigenfun...

A notion of Paley-Wiener spaces on combinatorial graphs is introduced. It is shown that functions from some of these spaces are uniquely determined by their values on some sets of vertices which are called the uniqueness sets. Such uniqueness sets are described in terms of Poincare-Wirtingertype inequalities. A reconstruction algorithm of Paley-Wiener functions from uniqueness sets which uses t...

We study the n-simplex nonconforming Crouzeix-Raviart element in approximating the n-dimensional second-order elliptic boundary value problems and the associated eigenvalue problems. By using the second Strang Lemma, optimal rate of convergence is established under the discrete energy norm. The error bound is also valid for the eigenfunction approximations. In addition, when eigenfunctions are ...

We establish an original result for the thermodynamic formalism in context of expanding circle transformations with indifferent fixed point. For observable whose modulus continuity is linked to dynamics near such a point, by identifying appropriate linear space evaluate action transfer operator, we show that there strictly positive eigenfunction associated maximal eigenvalue given as exponentia...

K e y w o r d s Radial basis functions, Eigenvalues, Numerical methods, Laplacian, corner singularities. 1. I N T R O D U C T I O N Many positive properties of radial basis function (RBF) methods have been identified in connection with boundary-value problems (BVPs) [1-4]. They are grid-free numerical schemes very suitable for problems in irregular geometries. They can exploit accurate and smoo...

The question of determining for which eigenvalues there exists an eigenfunction has the same number nodal domains as label associated eigenvalue (Courant-sharp property) was motivated by analysis minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, \ldots . A natural toy model further investigations is M\"...

We consider a class of diffusions controlled through the drift and jump size, driven by Lévy process nondegenerate Wiener process, we study infinite horizon (ergodic) risk-sensitive control problems for this model. start with Dirichlet eigenvalue problem in smooth bounded domains, which also allows us to generalize current results literature on exit rate problems. Then average minimization maxi...