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...

In this paper, the solution expansion for an inverted pendulum system with time delay is studied. The linearized model of this nonlinear system near its equilibrium is derived on the assumption that a unique equilibrium exists in it. Then the asymptotic expressions of its eigenvalues and the eigenvalues’ corresponding eigenvectors are obtained. Moreover, although the set of these eigenvectors d...

In this paper the concept of eigenvalues and eigenvectors of nonlinear systems, both continuousand discrete-time, is suggested. It represents a generalization of the concept known from linear control theory. Some basic properties, like invariance of eigenvalues under a (nonlinear) change of coordinates, possibility to transform the system to the diagonal form and, respectively, to the feedforwa...

In this paper a new simplified method of stability study dynamical nonlinear systems is proposed as an alternative to using Lyapunov’s method. Like the Lyapunov theorem, concept describes sufficient condition for be globally stable. The based on assumption that, not only state matrix contains information systems, but also eigenvectors. So, first we will write model in state-space representation...

Under rotation-equivariant decision theory, sample covariance matrix eigenvalues can be optimally shrunk by recombining eigenvectors with a (potentially nonlinear) function of the unobservable population matrix. The optimal shape this reflects loss/risk that is to minimized. We solve problem estimation under variety loss functions motivated statistical precedent, probability and differential ge...

This paper discusses the use of absolutely one-homogeneous regularization functionals in a variational, scale space, and inverse scale space setting to define a nonlinear spectral decomposition of input data. We present several theoretical results that explain the relation between the different definitions. Additionally, results on the orthogonality of the decomposition, a Parsevaltype identity...