We prove that a locally nitely dimensional shift-invariant linear space of distributions must be a linear subspace of some shift-invariant space generated by nitely many compactly supported distributions. If the locally nitely dimensional shift-invariant space is a subspace of the HH older continuous space C or the fractional

Suppose a smooth dynamical system has an invariant subspace and a parameter that leaves the dynamics in the invariant subspace invariant while changing the normal dynamics. Then we say the parameter is a normal parameter, and much is understood of how attractors can change with normal parameters. Unfortunately, normal parameters do not arise very often in practise. We consider the behaviour of ...

It is shown that, for any given polynomially normal matrix with respect to an indefinite inner product, a nonnegative (with respect to the indefinite inner product) invariant subspace always admits an extension to an invariant maximal nonnegative subspace. Such an extension property is known to hold true for general normal matrices if the nonnegative invariant subspace is actually neutral. An e...

It is shown that, for any given polynomially normal matrix with respect to an indefinite inner product, a nonnegative (with respect to the indefinite inner product) invariant subspace always admits an extension to an invariant maximal nonnegative subspace. Such an extension property is known to hold true for general normal matrices if the nonnegative invariant subspace is actually neutral. An e...

The invariant subspace method is used to classify a class of systems of nonlinear dispersive evolution equations and determine their invariant subspaces and exact solutions. A crucial step is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that systems of evolution equations admit. A few examples of presenting exact solutions with generalized sepa...

We discuss the reproducing kernel structure in shift-invariant spaces and the weighted shift-invariant spaces, and obtain the reconstruction formula in time-warped weighted shift-invariant spaces, then apply them to a spline subspace. In the spline subspace, we give a reconstruction formula in a time-warped spline subspace. 1. Introduction. The problem of reconstruction of a function f has been...

Nonlinear dynamical systems possessing reflection symmetry have an invariant subspace in the phase space. The dynamics within the invariant subspace can be random or chaotic. As a system parameter changes, the motion transverse to the invariant subspace can lose stability, leading to on-off intermittency. Under certain conditions, the bursting behavior is symmetry breaking. We demonstrate the p...

When a dynamical system possesses certain symmetry, there can be an invariant subspace in the phase space. In the invariant subspace there can be a chaotic attractor. As a parameter changes through a critical value, the chaotic attractor can lose stability with respect to perturbations transverse to the invariant subspace. We show that the loss of the transverse stability can lead to a symmetry...

In this paper, to solve the invariant subspace problem, contraction operators are classified into three classes ; (Case 1) completely nonunitary contractions with a non-trivial algebraic element, (Case 2) completely non-unitary contractions without a non-trivial algebraic element, or (Case 3) contractions which are not completely non-unitary. We know that every operator of (Case 3) has a non-tr...

We prove that a locally finite dimensional shift-invariant linear space of distributions must be a linear subspace of some shift-invariant space generated by finitely many compactly supported distributions. If the locally finite dimensional shift-invariant space is a subspace of the Hölder continuous space C or the fractional Sobolev space L , then the superspace can be chosen to be C or L , re...