Réflexivité d'une extension d'un opérateur sous-normal par un opérateur algébrique

A bounded linear operator on a Hilbert space is said to be reflexive if the operators which leave invariant the invariant subspaces of T are wot-limits of polynomials in T. In this paper we give a necessary and sufficient condition for an extension of a subnormal operator by an algebraic one to be reflexive.We also give a formula for the reflexivity defect of such extensions.     

General Approach to Integrating Invertible Dynamical Systems Defined by Transformations from the Cremona group Cr(P k n ) of Birational Transformations

A general approach is developed for integrating an invertible dynamical system defined by the composition of two involutions, i.e., a nonlinear one which is a standard Cremona transformation, and a linear one. By the Noether theorem, the integration of these systems is the foundation for integrating a broad class of Cremona dynamical systems. We obtain a functional equation for invariant homogeneous polynomials and sufficient conditions for the algebraic integrability of the systems under consideration. It is proved that Siegel's linearization theorem is applicable if the eigenvalues of the map at a fixed point are algebraic numbers.     

Milnor invariants for spatial graphs

Link-homotopy has been an active area of research for knot theorists since its introduction by Milnor in the 1950s. We introduce a new equivalence relation on spatial graphs called component-homotopy, which reduces to link-homotopy in the classical case. Unlike previous attempts at generalizing link-homotopy to spatial graphs, our new relation allows analogues of some standard link-homotopy results and invariants.In particular we can define a type of Milnor group for a spatial graph under component-homotopy, and this group determines whether or not the spatial graph is splittable. More surprisingly, we will also show that whether the spatial graph is splittable up to component-homotopy depends only on the link-homotopy class of the links contained within it. Numerical invariants of the relation will also be produced.     

Invariant means,right ideals and the structure of semitopological semigroups

A number of functional analytic properties associated with the existence of a right invariant mean on a function spaceF over a semitopological semigroup S are given. We give topological and algebraic conditions on a right ideal I of S for which a classF(S) of functions on S has a right invariant mean if and only if the classF(I) has a right invariant mean.     

Localization of periodic orbits of the Rössler system under variation of its parameters

The localization problem of compact invariant sets of the Rössler system is considered in this paper. The main interest is attracted to a localization of periodic orbits. We establish a number of algebraic conditions imposed on parameters under which the Rössler system has no compact invariant sets contained in half-spaces z > 0; z < 0 and in some others. We prove that if parameters (a, b, c) of the Rössler system are such that this system has no equilibrium points then it has no periodic orbits as well. In addition, we give localization conditions of compact invariant sets by using linear functions and one quadratic function.     

On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves

We give an upper bound for the maximum number N of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has exactly k nonsingular irreducible invariant algebraic curves. Additionally we provide sufficient conditions in order that all the algebraic limit cycles are hyperbolic. We also provide lower bounds for N.     

HOMOCLINIC CYCLES OF A QUADRATIC SYSTEM DESCRIBED BY QUARTIC CURVES

This paper is devoted to discussing the topological classification of the quartic invariant algebraic curves for a quadratic system. We obtain sufficient and necessary conditions which ensure that the homoclinic cycle of the system is defined by the quartic invariant algebraic curve. Finally, the corresponding global phase diagrams are drawn.     

Topology of Maximally Writhed Real Algebraic Knots

Oleg Viro introduced an invariant of rigid isotopy for real algebraic knots in ??³ which can be viewed as a first order Vassiliev invariant. In this paper we look at real algebraic knots of degree d with the maximal possible value of this invariant. We show that for a given d all such knots are topologically isotopic and explicitly identify their knot type.     

A Canonical Compatible Metric for Geometric Structures on Nilmanifolds

Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the ‘invariant Ricci’ flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish two geometric structures with Riemannian data, giving rise to a great deal of invariants.Our approach proposes to vary Lie brackets rather than inner products; our tool is the moment map for the action of a reductive Lie group on the algebraic variety of all Lie algebras, which we show to coincide in this setting with the Ricci operator. This gives us the possibility to use strong results from geometric invariant theory.Communicated by: Nigel Hitchin (Oxford) Mathematics Subject Classifications (2000): Primary: 53D05, 53D55; Secondary: 22E25, 53D20, 14L24, 53C30.     

Prooving Identities with Computer-Algebra -- Example: Algebraic Time-Derivative Estimation

For the case of continuous–time systems, this note contributes a detailed proof relating the so–called algebraic approach to time–derivative estimation, as proposed by Fliess and co–workers, to classical results from linear estimation theory. The proof is based on a modern computer–algebra proof technique that, in the main, resorts to the celebrated algorithm by Wilf and Zeilberger in the multisum case. As a result of the proof, the algebraic approach to time–derivative estimation is traced back, equivalently, to state estimation using the reconstructibility Gramian of the dynamic system, here, with respect to a particular nilpotent time–invariant input–free linear system. Additionally, the close relationship of the algebraic approach with least–squares time–derivative estimation is pointed out. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Invariant imbedding and the solution of Fredholm integral equations with displacement Kernels— comparative numerical experiments

This paper compares the relative efficiencies of the invariant imbedding method with the traditional solution techniques of successive approximations (Picard method), linear algebraic equations, and Sokolov's method of averaging functional corrections in solving numerically two representatives of a class of Fredholm integral equations. The criterion of efficiency is the amount of computing time necessary to obtain the solution to a specified degree of accuracy. The results of this computational investigation indicate that invariant imbedding has definite numerical advantages; more information was obtained in the same length of time as with the other methods, or even in less time. The conclusion emphasized is that a routine application of invariant imbedding may be expected to be computationally competitive with, if not superior to, a routine application of other methods for the solution of some classes of Fredholm integral equations.     

Larson理想中算子的插值

设a为nest代数，R^∞为A的Larson理想，R(A)为A的根，[R(A)]^s为R（A）的强拓扑闭。在文本中，我们给出[R(A)]^s的纯代数构造；且引进了一个新算子集合I，并证明了：若A的不变子空间格为几乎原子的，则R^∞＝[R(A)]^s=I。利用上述结果，我们研究了当A的不变子空间格为几乎原子时的Larson理想中的算子插值问题。我们得到算子方程AX＝Y在R^∞中有解A的充分必要条件。     

The 16th Hilbert problem on algebraic limit cycles

For real planar polynomial differential systems there appeared a simple version of the 16th Hilbert problem on algebraic limit cycles: Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m? In [J. Llibre, R. Ramírez, N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations 248 (2010) 1401-1409] Llibre, Ramírez and Sadovskaia solved the problem, providing an exact upper bound, in the case of invariant algebraic curves generic for the vector fields, and they posed the following conjecture: Is1+(m−1)(m−2)/2the maximal number of algebraic limit cycles that a polynomial vector field of degree m can have?In this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves. This result includes the Llibre et al.?s as a special one. For the polynomial vector fields having only non-dicritical invariant algebraic curves we answer the simple version of the 16th Hilbert problem.     

Invariante Zwischenkörper endlicher Körpererweiterungen

Let KM be a finite field extension. An intermediate field L is called invariant if there is an affine algebraic K-group acting on M with L as its invariant field. The question, which intermediate fields are invariant, was studied by Bégueri [1] for purely inseparable extensions and by Sweedler [6] for arbitrary extensions, but only for a restricted class of groups. In this paper Bégueri's result is generalized to arbitrary field extensions. Additionally it is shown that one can check whether a given intermediate field is invariant or not by computing the rank of certain matrices. As an application we get a class of invariant intermediate fields.     

A Natural Extension of a Left Invariant Lower Semi–Continuous Weight

In this paper, we describe a natural method to extend left invariant weights on C*–algebraic quantum groups. This method is then used to improve the invariance property of a left invariant weight. We also prove some kind of uniqueness result for left Haar weights on C*–algebraic quantum groups arising from algebraic ones.     

Lie isomorphisms of reflexive algebras

A Lie isomorphism ? between algebras is called trivial if ?=ψ+τ, where ψ is an (algebraic) isomorphism or a negative of an (algebraic) anti-isomorphism, and τ is a linear map with image in the center vanishing on each commutator. In this paper, we investigate the conditions for the triviality of Lie isomorphisms from reflexive algebras with completely distributive and commutative lattices (CDCSL). In particular, we prove that a Lie isomorphism between irreducible CDCSL algebras is trivial if and only if it preserves I-idempotent operators (the sum of an idempotent and a scalar multiple of the identity) in both directions. We also prove the triviality of each Lie isomorphism from a CDCSL algebra onto a CSL algebra which has a comparable invariant projection with rank and corank not one. Some examples of Lie isomorphisms are presented to show the sharpness of the conditions.     

Numerical Integration of Differential Algebraic Systems and Invariant Manifolds

The dynamics of a differential algebraic equation takes place on a lower dimensional manifold in phase space. Applying a numerical integration scheme, it is natural to ask if and how this geometric property is preserved by the discrete dynamical system. In the index-1 case answers to this question are obtained from the singularly perturbed case treated by Nipp and Stoffer, Numer. Math. 70 (1995), 245–257, for Runge-Kutta methods and in K. Nipp and D. Stoffer, Numer. Math. 74 (1996), 305–323, for linear multistep methods. As main result of this paper it is shown that also for Runge-Kutta methods and linear multistep methods applied to a index-2 problem of Hessenberg form there is a (attractive) invariant manifold for the discrete dynamical system and this manifold is close to the manifold of the differential algebraic equation.     

On planar zero-diagonal and zero-trace iterated maps

We call an iterated map zero-diagonal, if it has a zero-diagonal Jacobi matrix for all x,y. Similarly, zero-trace iterated maps are the maps with zero-trace Jacobi matrix. In this paper, we present some of the geometric and algebraic properties of zero-diagonal planar maps. However, the main focus of this paper is the analysis of the zero-trace planar maps by linear transforming them to a zero-diagonal ones. Some sufficient conditions for the transformation are obtained. Stability for non-hyperbolic fixed points, two types of codim-2 bifurcations, and the local/global invariant manifolds for zero-diagonal and zero-trace maps are investigated.     

Algebraic Torsion in Contact Manifolds

We extract an invariant taking values in \mathbbNè{￥}{\mathbb{N}\cup\{\infty\}} , which we call the order of algebraic torsion, from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic torsion of order 0 if and only if it is algebraically overtwisted (i.e. has trivial contact homology), and any contact 3-manifold with positive Giroux torsion has algebraic torsion of order 1 (though the converse is not true). We also construct examples for each k ? \mathbbN{k \in \mathbb{N}} of contact 3-manifolds that have algebraic torsion of order k but not k − 1, and derive consequences for contact surgeries on such manifolds.     

Geometric control of hybrid systems

In this paper, we present a geometric approach for computing controlled invariant sets for hybrid control systems. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support functions.