Gevrey normal forms of vector fields with one zero eigenvalue

We study normal forms of isolated singularities of vector fields in Rn or Cn. When all eigenvalues of the linear part of the vector field are nonzero, one can eliminate all so-called nonresonant terms from the equation provided some spectral condition (like Siegel) is satisfied. In this paper, we discuss the case where there is one zero eigenvalue (in that case Siegel's condition is not satisfied), and show that the formal normalizing transformations are either convergent or divergent of at most Gevrey type. In some cases, we show the summability of the normalizing transformations, which leads to the existence of analytic normal forms in complex sectors around the singularity.     

Non-deterministic ideal operators: An adequate tool for formalization in Data Bases

In this paper, we propose the application of formal methods to Software Engineering. The most used data model is the relational model and we present, within the general framework of lattice theory, this analysis of functional dependencies. For this reason, we characterize the concept of f-family by means of a new concept which we call non-deterministic ideal operator (nd.ideal-o). The study of nd.ideal-o.s allows us to obtain results about functional dependencies as trivial particularizations, to clarify the semantics of the functional dependencies and to progress in their efficient use, and to extend the concept of schema. Moreover, the algebraic characterization of the concept of Key of a schema allows us to propose new formal definitions in the lattice framework for classical normal forms in relation schemata. We give a formal definition of the normal forms for functional dependencies more frequently used in the bibliography: the second normal form (2FN), the third normal form(3FN) and Boyce-Codd's normal form (FNBC).     

Further reduction of normal forms of formal maps

Further reduction for classical normal forms of formal maps is considered in this note. Based on a recursive formula for computing the transformed map of a formal map under a near identity formal transformation, we develop the concepts of Nth order normal forms and infinite order normal forms for formal maps, and give some sufficient conditions for uniqueness of normal forms of formal maps. To cite this article: D. Wang et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Polynomial normal forms with exponentially small remainder for analytic vector fields

A key tool in the study of the dynamics of vector fields near an equilibrium point is the theory of normal forms, invented by Poincaré, which gives simple forms to which a vector field can be reduced close to the equilibrium. In the class of formal vector valued vector fields the problem can be easily solved, whereas in the class of analytic vector fields divergence of the power series giving the normalizing transformation generally occurs. Nevertheless the study of the dynamics in a neighborhood of the origin can very often be carried out via a normalization up to finite order. This paper is devoted to the problem of optimal truncation of normal forms for analytic vector fields in R^m. More precisely we prove that for any vector field in R^m admitting the origin as a fixed point with a semi-simple linearization, the order of the normal form can be optimized so that the remainder is exponentially small. We also give several examples of non-semi-simple linearization for which this result is still true.     

The Analytic and Formal Normal Form for the Nilpotent Singularity

We study orbital normal forms for analytic planar vector fields with nilpotent singularity. We show that the Takens normal form is analytic. In the case of generalized cusp we present the complete formal orbital normal form; it contains functional moduli. We interprete the coefficients of these moduli in terms of the hidden holonomy group.     

On the γ-filtration of oriented cohomology of complete spin-flags

We study the characteristic map of algebraic oriented cohomology of complete spin-flags and the ideal of invariants of formal group algebra. As an application, we provide an annihilator of the torsion part of the γ-filtration. Moreover, if the formal group law determined by the oriented cohomology theory is congruent to the additive formal group law modulo 2, then at degree 2 and 3, we show that the γ-filtration of complete spin-flags is torsion free.     

Essential variables and positions in terms

The paper deals with Σ-composition of terms, which allows us to extend the derivation rules in formal deduction of identities. The concepts of essential variables and essential positions of terms with respect to a set of identities form a key step in the simplification of the process of formal deduction. Σ-composition of terms is defined as replacement between Σ-equal terms. This composition induces ΣR-deductively closed sets of identities. In analogy to balanced identities we introduce and investigate Σ-balanced identities for a given set of identities Σ.     

An introduction to second degree forms

We are dealing with orthogonal sequences with respect to forms verifying a second degreee equation, i.e. that its formal Stieltjes functionS(u)(z) satisfies a quadratic equation of the formB(z)S ²(u)(z)+C(z)S(u)(z)+D(z)=0, whereB, C, D are polynomials. Various algebraic properties are given, especially those concerning the quasi-orthogonality of associated sequences. A classification is outlined. Some examples are quoted. In particular, we give the representation of Tchebychev co-recursive forms for any complex value of the parameter.     

Examples of non-conjugated holomorphic vector fields and foliations

We consider germs of holomorphic vector fields near the origin of with a saddle-node singularity, and the induced singular foliations. In a previous article we described the invariants addressing the analytical classification of these vector fields. They split into three parts: a formal, an orbital and a tangential component. For a fixed formal class, the orbital invariant (associated to the foliation) was obtained by Martinet and Ramis; we give it an integral representation. We then derive examples of non-orbitally conjugated foliations by the use of a “first-step” normal form, whose first-significative jet is an invariant. The tangential invariant also admits an integral representation, hence we derive explicit examples of vector fields, inducing the same foliation, that are not mutually conjugated. In addition, we provide a family of normal forms for vector fields orbitally equivalent to the model of Poincaré-Dulac.     

A note on strategy elimination in bimatrix games

In bimatrix games we study the process of successively eliminating strategies which are dominated by other pure strategies. We show how to perform this simplification in O(n³) time, where n is the number of strategies. We also prove that the problem is P-complete, which suggests that it is inherently sequential.     

First integrals of local analytic differential systems

We investigate formal and analytic first integrals of local analytic ordinary differential equations near a stationary point. A natural approach is via the Poincaré–Dulac normal forms: If there exists a formal first integral for a system in normal form then it is also a first integral for the semisimple part of the linearization, which may be seen as “conserved” by the normal form. We discuss the maximal setting in which all such first integrals are conserved, and show that all first integrals are conserved for certain classes of reversible systems. Moreover we investigate the case of linearization with zero eigenvalues, and we consider a three-dimensional generalization of the quadratic Dulac–Frommer center problem.     

On a Problem Related to Vortex Nucleation of Superconductivity

In this paper we study a singularly perturbed nonlinear partial differential system which arises in the mathematical theory of superheating field of superconductivity. We prove that the maximum points of the magnitude of solutions are located near the minimum point of curvature of domain boundary. This verifies rigorously a result of Chapman obtained by formal analysis regarding the location of the vortex nucleation. We also show that the solutions exhibit boundary layers.     

Systèmes dynamiques et lois de groupe formel sur les corps finis

Imitating the Lubin's philosophy on nonarchimedean dynamical systems, we prove that every finite family of inversibles series of Fq[[X]] which commute for the law ○ is connected with a finite family of automorphisms of a formal group over Fq. In some cases these formal groups are reduction on Fq of Lubin-Tate formal groups over finite extensions of Qp.     

Embedding smooth and formal diffeomorphisms through the Jordan-Chevalley decomposition

In [Xiang Zhang, The embedding flows of C^∞ hyperbolic diffeomorphisms, J. Differential Equations 250 (5) (2011) 2283-2298] Zhang proved that any local smooth hyperbolic diffeomorphism whose eigenvalues are weakly nonresonant is embedded in the flow of a smooth vector field. We present a new and more conceptual proof of such result using the Jordan-Chevalley decomposition in algebraic groups and the properties of the exponential operator.We characterize the hyperbolic smooth (resp. formal) diffeomorphisms that are embedded in a smooth (resp. formal) flow. We introduce a criterion showing that the presence of weak resonances for a diffeomorphism plus two natural conditions imply that it is not embeddable. This solves a conjecture of Zhang. The criterion is optimal, we provide a method to construct embeddable diffeomorphisms with weak resonances if we remove any of the conditions.     

Local analytic reduction of families of diffeomorphisms

We study local analytic simplification of families of analytic maps near a hyperbolic fixed point. A particularly important application of the main result concerns families of hyperbolic saddles, where Siegel's theorem is too fragile, at least in the analytic category. By relaxing on the formal normal form we obtain analytic conjugacies. Since we consider families, it is more convenient to state some results for analytic maps on a Banach space; this gives no extra complications. As an example we treat a family passing through a 1:−1 resonant saddle.     

Streaming Algorithms for Line Simplification

We study the following variant of the well-known line-simplification problem: we are getting a (possibly infinite) sequence of points p ₀,p ₁,p ₂,… in the plane defining a polygonal path, and as we receive the points, we wish to maintain a simplification of the path seen so far. We study this problem in a streaming setting, where we only have a limited amount of storage, so that we cannot store all the points. We analyze the competitive ratio of our algorithms, allowing resource augmentation: we let our algorithm maintain a simplification with 2k (internal) points and compare the error of our simplification to the error of the optimal simplification with k points. We obtain the algorithms with O(1) competitive ratio for three cases: convex paths, where the error is measured using the Hausdorff distance (or Fréchet distance), xy-monotone paths, where the error is measured using the Hausdorff distance (or Fréchet distance), and general paths, where the error is measured using the Fréchet distance. In the first case the algorithm needs O(k) additional storage, and in the latter two cases the algorithm needs O(k ²) additional storage.     

A generalized normal form and formal equivalence of two-dimensional systems with quadratic zero approximation: IV

We continue the study of invertible formal transformations of two-dimensional autonomous systems of differential equations with zero approximation represented by homogeneous polynomials of degree 2 and with perturbations in the form of power series without terms of order < 3. In the regular case, we consider systems that have the canonical form (αx ₁ ² ? sgnα x ₂ ² , x ₁ x ₂) with α ≠ 0 as the zero approximation. For such systems, we obtain resonance equations in closed form and use them to prove the theorem on the formal equivalence of systems and establish a generalized normal form to which any original system can be reduced by an invertible change of variables.     

Poincaré duality algebras mod two

We study Poincaré duality algebras over the field F₂ of two elements. After introducing a connected sum operation for such algebras we compute the corresponding Grothendieck group of surface algebras (i.e., Poincaré algebras of formal dimension 2). We show that the corresponding group for 3-folds (i.e., algebras of formal dimension 3) is not finitely generated, but does have a Krull-Schmidt property.We then examine the isomorphism classes of 3-folds with at most three generators of degree 3, provide a complete classification, settle which such occur as the cohomology of a smooth 3-manifold, and list separating invariants.The closing section and Appendix A provide several different means of constructing connected sum indecomposable 3-folds.     

Is the new ECTS system better than the traditional one? An application to the ECTS pilot-project at the University Pablo de Olavide

At the meeting of the joint Bologna Declaration, EU representatives agreed on the establishment of a common European Higher Education Area by 2010. Since then, several universities have implemented pilot projects, although no formal research has been carried out to analyse their results. In this study, we analysed one of these pilot-projects with two objectives. First, we examined the performance of the new system as compared to that of the traditional system. We used a procedure based on a modified model of Data Envelopment Analysis that is able to distinguish students’ efficiency (managerial efficiency) from efficiency based on the educational programme used (programme efficiency). Then we analysed whether the different systems perform differently for different types of students.