Persistence of lower-dimensional hyperbolic invariant tori for generalized Hamiltonian systems

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odd-dimensional. Under Rüssmann-type non-degenerate condition, by introducing a modified linear KAM iterative scheme, we proved that the majority of the lower-dimensional hyperbolic invariant tori persist under small perturbations for generalized Hamiltonian systems.     

Analytic intermediate dimensional elliptic tori for the planetary many-body problem

In our context,the planetary many-body problem consists of studying the motion of(n+1)-bodies under the mutual attraction of gravitation,where n planets move around a massive central body,the Sun.We establish the existence of real analytic lower dimensional elliptic invariant tori with intermediate dimension N lies between n and 3n-1 for the spatial planetary many-body problem.Based on a degenerate KolmogorovArnold-Moser(abbr.KAM)theorem proved by Bambusi et al.(2011),Berti and Biasco(2011),we manage to handle the difficulties caused by the degeneracy of this real analytic system.     

A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results

In this paper we prove rigorous results on persistence of invariant tori and their whiskers. The proofs are based on the parameterization method of [X. Cabré, E. Fontich, R. de la Llave, The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J. 52 (2) (2003) 283-328; X. Cabré, E. Fontich, R. de la Llave, The parameterization method for invariant manifolds. II. Regularity with respect to parameters, Indiana Univ. Math. J. 52 (2) (2003) 329-360]. The invariant manifolds results proved here include as particular cases of the usual (strong) stable and (strong) unstable manifolds, but also include other non-resonant manifolds. The method lends itself to numerical implementations whose analysis and implementation is studied in [A. Haro, R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms, preprint, 2005; A. Haro, R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical implementation and examples, preprint, 2005]. The results are stated as a posteriori results. Namely, that if one has an approximate solution which is not degenerate, then, one has a true solution not too far from the approximate one. This can be used to validate the results of numerical computations.     

Two-dimensional invariant tori in the neighborhood of an elliptic equilibrium of Hamiltonian systems

In this paper, we prove that a Hamiltonian system possesses either a four-dimensional invaxiant disc or an invariant Cantor set with positive （n ＋ 2）-dimensional Lebesgue measure in the neighborhood of an elliptic equilibrium provided that its lineaxized system at the equilibrium satisfies some small divisor conditions. Both of the invariant sets are foliated by two-dimensional invaxiant tori carrying quasi-oeriodic solutions.     

On the existence of invariant tori in nearly-integrable Hamiltonian systems with finitely differentiable perturbations

We prove the existence of invariant tori in Hamiltonian systems, which are analytic and integrable except a 2n-times continuously differentiable perturbation (n denotes the number of the degrees of freedom), provided that the moduli of continuity of the 2n-th partial derivatives of the perturbation satisfy a condition of finiteness (condition on an integral), which is more general than a Hölder condition. So far the existence of invariant tori could be proven only under the condition that the 2n-th partial derivatives of the perturbation are Hölder continuous.     

Conservation and bifurcation of an invariant torus of a vector field

We consider small perturbations with respect to a small parameter ε≥0 of a smooth vector field in ℝ^n+m possessing an invariant torusT ^m. The flow on the torusT ^m is assumed to be quasiperiodic withm basic frequencies satisfying certain conditions of Diophantine type; the matrix Ω of the variational equation with respect to the invariant torus is assumed to be constant. We investigate the existence problem for invariant tori of different dimensions for the case in which Ω is a nonsingular matrix that can have purely imaginary eigenvalues. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 34–44, January, 1997. Translated by S. K. Lando     

Non-surjectivity of the Clifford invariant map

The question whether there exists a commutative ringA for which there is an element in the 2-torsion of the Brauer group not represented by a Clifford algebra was raised by Alex Hahn. Such an example is constructed in this paper and is arrived at using certain results of Parimala-Sridharan and Parimala-Scharlau which are also reviewed here. Dedicated to the memory of Professor K G Ramanathan     

Hopf invariants, Toda brackets and the reduced diagonal map

That there is a close connection between crude (or suspended) Hopf invariants and the reduced diagonal map has long been known. In this paper we build on a classical result of Boardman and Steer (1967) to explore some further relationships in terms of conic structures. In a related result we show that certain Toda brackets contain crude Hopf invariants as elements. In particular, using this latter observation, detection of crude Hopf invariants can be carried out.     

ADIABATIC INVARIANTS OF SLOWLY VARYING THREE－DIMENSIONAL SYSTEMS AND EXISTENCE OF INVARIANT TORI OF LOTKA－VOLTERRA EQUATION

ＡＤＩＡＢＡＴＩＣＩＮＶＡＲＩＡＮＴＳＯＦＳＬＯＷＬＹＶＡＲＹＩＮＧＴＨＲＥＥ－ＤＩＭＥＮＳＩＯＮＡＬＳＹＳＴＥＭＳＡＮＤＥＸＩＳＴＥＮＣＥＯＦＩＮＶＡＲＩＡＮＴＴＯＲＩＯＦＬＯＴＫＡ－ＶＯＬＴＥＲＲＡＥＱＵＡＴＩＯＮＬＩＪＩＢＩＮＺＨＡＯＸＩＡＯＨ...     

CONSTRUCTIONS OF THE INVARIANT SETS AND INVARIANT MEASURES

In this paper, we will discuss the constructiOn problems about the invariant sets and invariant measures of continues maps~ which map complexes into themselves, using simplical approximation and Markov cbeirs. In [7], the author defined a matrix by using r-normal subdivi of the w,dimensional unit cube, considered it a Markov matrix, and constructed the invariantset and invafiant measure, In fact, the matrix he defined is not Markov matrix generally. So wewill give [7] and amendment in the last pert of this paper. We also construct an invariant set thatis the chain-recurrent set of the map by means of a non-negative matrix which only depends on themap. At hst, we will prove the higher dimension?Banach variation formuls that can simplify thetransition matrix.     

A geometrical method for the approximation of invariant tori

We consider a numerical method based on the so-called “orthogonality condition” for the approximation and continuation of invariant tori under flows. The basic method was originally introduced by Moore [Computation and parameterization of invariant curves and tori, SIAM J. Numer. Anal. 15 (1991) 245–263], but that work contained no stability or consistency results. We show that the method is unconditionally stable and consistent in the special case of a periodic orbit. However, we also show that the method is unstable for two-dimensional tori in three-dimensional space when the discretization includes even numbers of points in both angular coordinates, and we point out potential difficulties when approximating invariant tori possessing additional invariant sub-manifolds (e.g., periodic orbits). We propose some remedies to these difficulties and give numerical results to highlight that the end method performs well for invariant tori of practical interest.     

A finite type invariant of order at most 4 for genus 2 handlebody-knots is a constant map

We show that a finite type invariant of order at most 4 for genus 2 handlebody-knots is a constant map. For this purpose, we give a concrete basis for the vector space of all finite type invariants of order at most 4 for spatial theta-curves, which makes a correction to Koike?s result. Each invariant of the basis is derived from the HOMFLYPT polynomial of the associated 3-component link of a spatial theta-curve.     

A refined invariant subspace method and applications to evolution equations

The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations.The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit.A two-component nonlinear system of dissipative equations is analyzed to shed light on the resulting theory,and two concrete examples are given to find invariant subspaces associated with 2nd-order and 3rd-order linear ordinary differential equations and their corresponding exact solutions with generalized separated variables.     

(u,02)-广义次似凸映射的择一定理及其对向量优化的应用

在序线性拓扑空间中,引入（u,02）-广义次似凸映射,建立了（u,02）-广义次似凸映射的一个择一定理.最后,利用此定理,在序线性拓扑空间中得到了带广义不等式约束的向量极值问题的最优性必要条件和充分条件.     

Topologies related to the prox map and the restricted center map

In this paper we review some of the recently introduced hypertopologies on collections of nonempty closed subsets of a metrizable space in the context of continuity properties of the prox map and the restricted center map. We identify here suitable families A, B of nonempty closed sets equipped with natural topologies, as coordinate spaces for the gap functional and the restricted radius functional, with a view to explore bivariate continuity of these functionals. This leads us to sharpenings of many known results as well as to some new results for upper semicontinuity of the prox map and the restricted center map. This, in turn, also leads us to improvements of certain best approximation results for convex-valued multifunctions and fixed point theorems for such multifunctions.     

A symmetry of sphere map implies its chaos*

A well-known example, given by Shub, shows that for any |d| ≥ 2 there is a self-map of the sphere Sⁿ, n ≥ 2, of degree d for which the set of non-wandering points consists of two points. It is natural to ask which additional assumptions guarantee an infinite number of periodic points of such a map. In this paper we show that if a continuous map f : Sⁿ → Sⁿ commutes with a free homeomorphism g : Sⁿ → Sⁿ of a finite order, then f has infinitely many minimal periods, and consequently infinitely many periodic points. In other words the assumption of the symmetry of f originates a kind of chaos. We also give an estimate of the number of periodic points. *Research supported by KBN grant nr 2 P03A 045 22.     

Invariant functionals and the uniqueness of invariant norms

Let τ be a representation of a compact group G on a Banach space (X,||·||). The question we address is whether X carries a unique invariant norm in the sense that ||·|| is the unique norm on X for which τ is a representation. We characterize the uniqueness of norm in terms of the automatic continuity of the invariant functionals in the case when X is a dual Banach space and τ is a -continuous representation of G on X such that τ(G) consists of -continuous operators. We illustrate the usefulness of this characterization by studying the uniqueness of the norm on the spaces Lp(Ω), where Ω is a locally compact Hausdorff space equipped with a positive Radon measure and G acts on Ω as a group of continuous invertible measure-preserving transformations.     

B(0,N)-graded Lie superalgebras coordinatized by quantum tori Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

We use a fermionic extension of the bosonic module to obtain a class of B(0, A)graded Lie superalgebras with nontrivial central extensions.     

Ketu and the second invariant of a quadratic space

Using the Chern classes defined by Grothendieck, we map aK-theoretic invariant of invertible symmetric matrices defined by Giffen to the Clifford invariant.According to Herrmann Graßman'sWörterbuch zum Rig-veda, Ketu may not be connected with K₂ but es bezeichnet das, was sich sichtbar oder kenntlich macht ...