Analytic invariant curves for an iterative equation related to dissipative standard map

In this paper, we study existence of invariant curves of an iterative equation which is from dissipative standard map. By constructing an invertible analytic solution g (x ) of an auxiliary equation of the form invertible analytic solutions of the form g (λ g  ^? 1(x )) for the original iterative functional equation are obtained. Besides the hyperbolic case 0 < |λ | < 1, we focus on those λ on the unit circle S ¹, that is, |λ | = 1. We discuss not only those λ at resonance, that is, at a root of the unity, but also those λ near resonance under the Brjuno condition. Copyright © 2016 John Wiley & Sons, Ltd.     

A symplectic map and its application to the persistence of lower dimensional invariant tori

We give a nonlinear symplectic coordinator transformation, which can move the normal frequencies of the lower dimensional torus up to (k,w) where ω is the frequency vector of the torus. That means the normal frequencies with a difference (k,w) may be regarded as the same. As an application, we derive a persistence result on lower dimensional tori of nearly integrable Hamiltonian systems when the second Melnikov’s condition is partially violated.     

Douglas algebras which are invariant under the Bourgain map

Part of this work was done while the second author was visiting the University of Georgia, Athens/Georgia, USA.     

Existence of an invariant form under a linear map

Let F be a field of characteristic different from 2 and V be a vector space over F. Let J: α → α ^J be a fixed involutory automorphism on F. In this paper we answer the following question: given an invertible linear map T: V → V, when does the vector space V admit a T-invariant nondegenerate J-hermitian, resp. J-skew-hermitian, form?     

Uniqueness of left invariant symplectic structures on the affine Lie group

We show the uniqueness of left invariant symplectic structures on the affine Lie group under the adjoint action of , by giving an explicit formula of the Pfaffian of the skew symmetric matrix naturally associated with , and also by giving an unexpected identity on it which relates two left invariant symplectic structures. As an application of this result, we classify maximum rank left invariant Poisson structures on the simple Lie groups and . This result is a generalization of Stolin's classification of constant solutions of the classical Yang-Baxter equation for and .

     

Estimates for the energy of a symplectic map

Supported by SFB 237 and Procope     

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     

A bound on the Castelnuovo-Mumford regularity for curves

Recall that a projective curve in with ideal sheaf is said to be n-regular if for every integer and that in this case, it is cut out scheme-theoretically by equations of degree at most n. The purpose here is to show that an irreducible, reduced, projective curve of degree d and large arithmetic genus satisfies a smaller regularity bound than the optimal one . For example, if then a curve is -regular unless it is embedded by a complete linear system of degree . Received: 29 May 2000 / Published online: 24 September 2001     

On the Castelnuovo-Mumford regularity of connected curves

In this paper we prove that the regularity of a connected curve is bounded by its degree minus its codimension plus 1. We also investigate the structure of connected curves for which this bound is optimal. In particular, we construct connected curves of arbitrarily high degree in having maximal regularity, but no extremal secants. We also show that any connected curve in of degree at least 5 with maximal regularity and no linear components has an extremal secant.

     

Orbit algebras that are invariant under stable equivalences of Morita type

In this note we show that there are a lot of orbit algebras that are invariant under stable equivalences of Morita type between self-injective algebras. There are also indicated some links between Auslander-Reiten periodicity of bimodules and noetherianity of their orbit algebras.     

On the existence of an invariant non-degenerate bilinear form under a linear map

Let V be a vector space over a field F. Assume that the characteristic of F is large, i.e. char(F)>dimV. Let T:V→V be an invertible linear map. We answer the following question in this paper. When doesVadmit a T-invariant non-degenerate symmetric (resp. skew-symmetric) bilinear form? We also answer the infinitesimal version of this question.Following Feit and Zuckerman 2, an element g in a group G is called real if it is conjugate in G to its own inverse. So it is important to characterize real elements in GL(V,F). As a consequence of the answers to the above question, we offer a characterization of the real elements in GL(V,F).Suppose V is equipped with a non-degenerate symmetric (resp. skew-symmetric) bilinear form B. Let S be an element in the isometry group I(V,B). A non-degenerate S-invariant subspace W of (V,B) is called orthogonally indecomposable with respect to S if it is not an orthogonal sum of proper S-invariant subspaces. We classify the orthogonally indecomposable subspaces. This problem is non-trivial for the unipotent elements in I(V,B). The level of a unipotent T is the least integer k such that (T-I)^k=0. We also classify the levels of unipotents in I(V,B).