Weak Non-degeneracy Condition Research Articles

Partial Preservation of Frequencies and Floquet Exponents of Invariant Tori in the Reversible KAM Context 2

We consider the persistence of smooth families of invariant tori in the reversible context 2 of KAM theory under various weak nondegeneracy conditions via Herman’s method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question. The nondegeneracy conditions we employ ensure the preservation of any prescribed subsets of the frequencies of the unperturbed tori and of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus).

Necessary conditions for two-weight inequalities for singular integral operators

We prove necessary conditions on pairs $(\mu ,\nu )$ of measures for a singular integral operator $T$ to satisfy weak $(p,p)$ inequalities, $1\leq p \lt \infty $, provided the kernel of $T$ satisfies a weak non-degeneracy condition first introduced by Ste

On the linearization of vector fields on a torus with prescribed frequency

In this paper we are mainly concerned with the linearization of the flow with prescribed frequency for analytic perturbation of constant vector fields on a torus under weaker non-degeneracy condition and non-resonant condition. As is well known the perturbation of constant vector fields may induce a shift of frequency, when Kolmogorov's non-degeneracy condition is violated. By introducing external parameters and using the polynomial structure to truncate, we prove that if the frequency mapping has the nonzero Brouwer's topological degree at some non-resonant frequency, then the conjugated vector fields will have a linear flow with this frequency.

Improved Convergence Properties of the Relaxation Schemes of Kadrani et al. and Kanzow and Schwartz for MPEC

This paper concentrates on improving the convergence properties of the relaxation schemes introduced by Kadrani et al. and Kanzow and Schwartz for mathematical program with equilibrium constraints (MPEC) by weakening the original constraint qualifications. It has been known that MPEC relaxed constant positive-linear dependence (MPEC-RCPLD) is a class of extremely weak constraint qualifications for MPEC, which can be strictly implied by MPEC relaxed constant rank constraint qualification (MPEC-RCRCQ) and MPEC relaxed constant positive-linear dependence (MPEC-rCPLD), of course also by the MPEC constant positive-linear dependence (MPEC-CPLD). We show that any accumulation point of stationary points of these two approximation problems is M-stationarity under the MPEC-RCPLD constraint qualification, and further show that the accumulation point can even be S-stationarity coupled with the asymptotically weak nondegeneracy condition.

Partial Preservation of Frequencies and Floquet Exponents of Invariant Tori in the Reversible KAM Context 2

We consider the persistence of smooth families of invariant tori in the reversible context 2 of KAM theory under various weak nondegeneracy conditions via Herman’s method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question. The nondegeneracy conditions we employ ensure the preservation of any prescribed subsets of the frequencies of the unperturbed tori and of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus).

Lens rigidity in scattering by non-trapping obstacles

We prove that if two non-trapping obstacles in \(\mathbb {R}^n\) satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling times or (almost) the same scattering length spectrum, then they coincide.

Linearization of a quasi-periodically forced flow on 𝕋m under Brjuno–Rüssmann non-resonant condition

In this paper, we prove that a class of quasi-periodically forced flow with external parameters on a m-torus is linearizable, i.e. conjugable to the quasi-periodic rotation, under some appropriate conditions. More concretely, utilizing Pöschel–Rüssmann KAM method we can find a quasi-periodic transformation such that this flow becomes a quasi-periodic linear flow in a Cantor subset of parameters when forcing frequency and unperturbed rotation vector to satisfy the Brjuno–Rüssmann’s non-resonant condition and a weaker non-degeneracy condition, respectively. While in the case where the unperturbed rotation vector does not satisfy the weaker non-degeneracy condition, Herman’s method is applied to overcome the degeneracy so that we show that some perturbed systems can conjugate to the original system with prescribed rotation vector Finally, as an application, our results are used to investigate the spectrum and eigenfunctions of a quasi-periodic Schrödinger operators.

On invariant tori of vector field under weaker non-degeneracy condition

In this paper we prove the persistence of invariant tori for analytic perturbation of constant vector field under weaker non-degeneracy condition. In the proof we introduce a parameter q and make the steps of KAM iteration infinitely small in the speed of function \({q^{n} \epsilon}\), \({0 < q < 1}\), rather than super exponential function.

On "Arnold's theorem" on the stability of the solar system

Arnold's theorem on the planetary problem states that, assuming that the masses of $n$ planets are small enough, there exists in the phase space a set of initial conditions of positive Lebesgue measure, leading to quasiperiodic motions with $3n-1$ frequencies. Arnold's initial proof is complete only for the plane $2$-planet problem. Arnold had missed a resonance later discovered by Herman. The first complete proof, by Herman-Féjoz, relies on the weak non-degeneracy condition of Arnold-Pyartli. A second proof, by Chierchia-Pinzari, is closer to Arnold's initial idea and shows the strong non-degeneracy of the problem after suitable reduction by (part of) the symmetry of rotation. We review and compare these proofs. In an appendix, we define the Poincaré coordinates and prove their symplectic nature through the shortest possible computation.

Degenerate KAM theory for partial differential equations

This paper deals with degenerate KAM theory for lower dimensional elliptic tori of infinite dimensional Hamiltonian systems, depending on one parameter only. We assume that the linear frequencies are analytic functions of the parameter, satisfy a weak non-degeneracy condition of Rüssmann type and an asymptotic behavior. An application to nonlinear wave equations is given.

Gevrey Regularity of Invariant Curves of Analytic Reversible Mappings

We prove the existence of a Gevrey family of invariant curves for analytic reversible mappings under weaker nondegeneracy condition. The index of the Gevrey smoothness of the family could be any number , where is the exponent in the small divisors condition and is the order of degeneracy of the reversible mappings. Moreover, we obtain a Gevrey normal form of the reversible mappings in a neighborhood of the union of the invariant curves.

A New Regularization Scheme for Mathematical Programs with Complementarity Constraints

We propose a new regularization scheme for mathematical programs with complementarity constraints (MPCC) by relaxing all the constraints of the complementarity system. We show that, under the MPCC-linear independence constraint qualifications (MPCC-LICQ), the Lagrange multipliers exist for this regularization. Our method has strong convergence properties under MPCC-linear independence constraint qualifications and some weak conditions of the strict complementarity. In particular, under MPCC-LICQ, it is shown that any accumulation point of the regularized stationary points is M-stationary for the MPCC problem, and if the asymptotically weak nondegeneracy condition holds at a stationary point of the regularized problem, then it is strongly stationary. An algorithm for solving the proposed regularization is presented and numerical experiments are reported. Some comparisons with other methods are discussed with illustrative examples.

Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method

We consider quasi-periodic (with $N$ basic frequencies) non-autonomous perturbations of Hamiltonian, reversible, volume preserving, and dissipative systems. The unperturbed systems possess analytic families of invariant $n$-tori carrying conditionally periodic motions, are allowed to depend on external parameters, and are assumed to satisfy just very weak nondegeneracy conditions. We construct invariant $(n+N)$-tori in perturbed systems following M.R. Herman's approach: additional external parameters are introduced to remove degeneracies and then are eliminated via an appropriate number-theoretical lemma concerning Diophantine approximations of dependent quantities.

Quadratic Convergence of a Nonsmooth Newton-Type Method for Semidefinite Programs Without Strict Complementarity

We consider a Newton-type method for the solution of semidefinite programs. This Newton-type method is based on a semismooth reformulation of the semidefinite program as a nonsmooth system of equations. We establish local quadratic convergence of this method under a linear independence assumption and a weak nondegeneracy condition that implies uniqueness of the optimal solution but does not imply strict complementarity.

Existence of Invariant Tori for Certain Weakly Reversible Mappings

In this paper, we consider certain mappings, \({\mathcal {M}}\) , sufficiently close to an integrable one, which is weakly reversible with respct to the mappings \({\mathcal {G}}\) sufficiently close to an involution of tye (m, n), where m, n∈Z + are arbitrary. Under some weak non-degeneracy condition, we construct a uniform KAM iteration for providing the existence of a Cantor family of m-tori invariant under the reversible mappings \({\mathcal {M}}\) and the reversing mapping \({\mathcal {G}}\).

Fast Solution of the Radial Basis Function Interpolation Equations: Domain Decomposition Methods

In this paper we consider domain decomposition methods for solving the radial basis function interpolation equations. There are three interwoven threads to the paper. The first thread provides good ways of setting up and solving small- to medium-sized radial basis function interpolation problems. These may occur as subproblems in a domain decomposition solution of a larger interpolation problem. The usual formulation of such a problem can suffer from an unfortunate scale dependence not intrinsic in the problem itself. This scale dependence occurs, for instance, when fitting polyharmonic splines in even dimensions. We present and analyze an alternative formulation, available for all strictly conditionally positive definite basic functions, which does not suffer from this drawback, at least for the very important example previously mentioned. This formulation changes the problem into one involving a strictly positive definite symmetric system, which can be easily and efficiently solved by Cholesky factorization. The second section considers a natural domain decomposition method for the interpolation equations and views it as an instance of von Neumann's alternating projection algorithm. Here the underlying Hilbert space is the reproducing kernel Hilbert space induced by the strictly conditionally positive definite basic function. We show that the domain decomposition method presented converges linearly under very weak nondegeneracy conditions on the possibly overlapping subdomains. The last section presents some algorithmic details and numerical results of a domain decomposition interpolatory code for polyharmonic splines in 2 and 3 dimensions. This code has solved problems with 5 million centers and can fit splines with 10,000 centers in approximately 7 seconds on very modest hardware.

On the number of lines in 4-dimensional linear spaces

Assuming a weak non-degeneracy condition, we show that a linear spaceL of dimension at least 4 withv=q4+q3+q2+q+1 points,q > 1 any positive real number, has at least (q2+1)v lines with equality if and only ifq is a prime power andL = PG(4,q).

A global condition for quasi-random behavior in a class of conservative systems

Devaney has shown that an autonomous Hamiltonian system in dimension 4, with an orbit homoclinic to a saddle-focus equilibrium, admits a chaotic behavior as soon as the homoclinic orbit is the transverse intersection of the stable and unstable manifolds. In this paper we deal with two classes of saddle-focus systems: Lagrangian systems defined on a two-manifold in the presence of a gyroscopic force, and fourth-order systems arising in water-wave theory. We first establish, by a standard variational method, the existence of a homoclinic orbit. Then, under a weak nondegeneracy condition, we show that it gives rise to an infinite family of multibump homoclinic solutions and that the dynamics are chaotic. Our condition is much easier to check than transversality. For example, it is automatically satisfied for gyroscopic systems on a two-torus, for topological reasons. © 1996 John Wiley & Sons, Inc.

The iteration-approximation decoupling in the reversible KAM theory.

General theorems on the persistence of quasiperiodic motions in reversible flows and diffeomorphisms satisfying very weak nondegeneracy conditions are obtained by a new method. The essence of this method is that the reversible system under consideration is embedded in a multiparameter family of reversible systems, and standard results on Diophantine approximations of dependent quantities are then applied to Whitney-smooth Cantor foliations of invariant tori of this family. Invariant tori are constructed for all the permissible values of m, p, q (for vector fields V) or m, p, q, P, Q (for diffeomorphisms A) where m is the torus dimension, (q,p) is the type of the reversing involution G, and (Q,P) is the type of the involution AG. The excitation of elliptic normal modes is also considered. (c) 1995 American Institute of Physics.

On the number of lines in planar spaces

SupposeS is a planar space withv>4 points and letq be the positive real number such thatv=q3+q2+q+1. Assuming a weak non-degeneracy condition, we shall show thatS has at least (q2+1)(q2+q+1) lines with equality iffq is a prime power andS=PG(3,q).